Research Article Paracomplex Paracontact Pseudo-Riemannian...

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Research Article Paracomplex Paracontact Pseudo-Riemannian Submersions S. S. Shukla and Uma Shankar Verma Department of Mathematics, University of Allahabad, Allahabad 211002, India Correspondence should be addressed to Uma Shankar Verma; [email protected] Received 25 February 2014; Accepted 7 April 2014; Published 7 May 2014 Academic Editor: Bennett Palmer Copyright © 2014 S. S. Shukla and U. S. Verma. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We introduce the notion of paracomplex paracontact pseudo-Riemannian submersions from almost para-Hermitian manifolds onto almost paracontact metric manifolds. We discuss the transference of structures on total manifolds and base manifolds and provide some examples. We also obtain the integrability condition of horizontal distribution and investigate curvature properties under such submersions. 1. Introduction e theory of Riemannian submersion was introduced by O’Neill [1, 2] and Gray [3]. It is known that the applications of such Riemannian submersion are extensively used in Kaluza- Klein theories [4, 5], Yang-Mill equations [6, 7], the theory of robotics [8], and supergravity and superstring theories [5, 9]. ere is detailed literature on the Riemannian submer- sion with suitable smooth surjective map followed by differ- ent conditions applied to total space and on the fibres of sur- jective map. e Riemannian submersions between almost Hermitian manifolds have been studied by Watson [10]. e Riemannian submersions between almost contact manifolds were studied by Chinea [11]. He also concluded that if is an almost Hermitian manifold with structure ( , ) and is an almost contact metric manifold with structure (, , , ), then there does not exist a Riemannian submersion : → which commutes with the structures on and ; that is, we cannot have the condition ∗ ∘ =∘ ∗ . Chinea also defined the Riemannian submersion between almost complex manifolds and almost contact manifolds and studied some properties and interrelations between them [12]. In [13], G¨ und¨ uzalp and Sahin gave the concept of paracontact paracomplex semi-Riemannian submersion between almost paracontact metric manifolds and almost para-Hermitian manifolds submersion giving an example and studied some geometric properties of such submersions. An almost paracontact structure on a differentiable man- ifold was introduced by Sato [14], which is an analogue of an almost contact structure and is closely related to almost product structure. An almost contact manifold is always odd dimensional but an almost paracontact manifold could be even dimensional as well. e paracomplex geometry has been studied since the first papers by Rashevskij [15], Libermann [16], and Patterson [17] until now, from several different points of view. e subject has applications to several topics such as negatively curved manifolds, mechanics, elliptic geometry, and pseudo- Riemannian space forms. Paracomplex and paracontact geometries are topics with many analogies and also with differences with complex and contact geometries. is motivated us to study the pseudo-Riemannian sub- mersion between pseudo-Riemannian manifolds equipped with paracomplex and paracontact structures. In this paper, we give the notion of paracomplex paracon- tact pseudo-Riemannian submersion between almost para- complex manifolds and almost paracontact pseudometric manifolds giving some examples and study the geometric properties and interrelations under such submersions. e composition of the paper is as follows. In Section 2, we collect some basic definitions, formulas, and results on almost paracomplex manifolds, almost paracontact pseu- dometric manifolds, and pseudo-Riemannian submersion. Hindawi Publishing Corporation Geometry Volume 2014, Article ID 616487, 12 pages http://dx.doi.org/10.1155/2014/616487

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Research ArticleParacomplex Paracontact Pseudo-Riemannian Submersions

S S Shukla and Uma Shankar Verma

Department of Mathematics University of Allahabad Allahabad 211002 India

Correspondence should be addressed to Uma Shankar Verma umashankarverma7gmailcom

Received 25 February 2014 Accepted 7 April 2014 Published 7 May 2014

Academic Editor Bennett Palmer

Copyright copy 2014 S S Shukla and U S VermaThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We introduce the notion of paracomplex paracontact pseudo-Riemannian submersions from almost para-Hermitian manifoldsonto almost paracontact metric manifolds We discuss the transference of structures on total manifolds and base manifolds andprovide some examples We also obtain the integrability condition of horizontal distribution and investigate curvature propertiesunder such submersions

1 Introduction

The theory of Riemannian submersion was introduced byOrsquoNeill [1 2] and Gray [3] It is known that the applications ofsuch Riemannian submersion are extensively used in Kaluza-Klein theories [4 5] Yang-Mill equations [6 7] the theory ofrobotics [8] and supergravity and superstring theories [5 9]

There is detailed literature on the Riemannian submer-sion with suitable smooth surjective map followed by differ-ent conditions applied to total space and on the fibres of sur-jective map The Riemannian submersions between almostHermitian manifolds have been studied by Watson [10] TheRiemannian submersions between almost contact manifoldswere studied by Chinea [11] He also concluded that if 119872 isan almost Hermitian manifold with structure (119869 119892) and119872 isan almost contact metric manifold with structure (120601 120585 120578 119892)then there does not exist a Riemannian submersion119891 119872 rarr

119872 which commutes with the structures on 119872 and 119872 thatis we cannot have the condition 119891

lowast∘ 119869 = 120601 ∘ 119891

lowast Chinea

also defined the Riemannian submersion between almostcomplexmanifolds and almost contactmanifolds and studiedsome properties and interrelations between them [12] In[13] Gunduzalp and Sahin gave the concept of paracontactparacomplex semi-Riemannian submersion between almostparacontact metric manifolds and almost para-Hermitianmanifolds submersion giving an example and studied somegeometric properties of such submersions

An almost paracontact structure on a differentiable man-ifold was introduced by Sato [14] which is an analogue ofan almost contact structure and is closely related to almostproduct structure An almost contact manifold is always odddimensional but an almost paracontact manifold could beeven dimensional as well

The paracomplex geometry has been studied since thefirst papers by Rashevskij [15] Libermann [16] and Patterson[17] until now from several different points of view Thesubject has applications to several topics such as negativelycurved manifolds mechanics elliptic geometry and pseudo-Riemannian space forms Paracomplex and paracontactgeometries are topics with many analogies and also withdifferences with complex and contact geometries

This motivated us to study the pseudo-Riemannian sub-mersion between pseudo-Riemannian manifolds equippedwith paracomplex and paracontact structures

In this paper we give the notion of paracomplex paracon-tact pseudo-Riemannian submersion between almost para-complex manifolds and almost paracontact pseudometricmanifolds giving some examples and study the geometricproperties and interrelations under such submersions

The composition of the paper is as follows In Section 2we collect some basic definitions formulas and results onalmost paracomplex manifolds almost paracontact pseu-dometric manifolds and pseudo-Riemannian submersion

Hindawi Publishing CorporationGeometryVolume 2014 Article ID 616487 12 pageshttpdxdoiorg1011552014616487

2 Geometry

In Section 3 we define paracomplex paracontact pseudo-Riemannian submersion giving some relevant examples andinvestigate transference of structures on the total manifoldsand base manifolds under such submersions In Section 4curvature relations between total manifolds base manifoldsand fibres are studied

2 Preliminaries

21 Almost Paracontact Manifolds Let 119872 be a (2119899 + 1)-dimensional Riemannianmanifold 120601 a (11)-type tensor field120585 a vector field called characteristic vector field and 120578 a 1-form on 119872 Then (120601 120585 120578) is called an almost paracontactstructure on119872 if

1206012

119883 = 119883 minus 120578 (119883) 120585 120578 (120585) = 1 (1)

and the tensor field 120601 induces an almost paracomplexstructure on the distributionD = ker(120578) [18 19]

119872 is said to be an almost paracontact manifold if it isequipped with an almost paracontact structure Again119872 iscalled an almost paracontact pseudometric manifold if it isendowed with a pseudo-Riemannian metric 119892 of signature(minus minus minus minus⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

(119899-times) + + + +⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

(119899+1)-times)) such that

119892 (120601119883 120601119884) = 119892 (119883 119884) minus 120576120578 (119883) 120578 (119884) forall119883 119884 isin Γ (119879119872)

(2)

where 120576 = 1 or minus1 according to the characteristic vector field120585 is spacelike or timelike It follows that

119892 (120585 120585) = 120576 (3)119892 (120585 119883) = 120576120578 (119883) (4)

119892 (119883 120601119884) = 119892 (120601119883 119884) forall119883 119884 isin Γ (119879119872) (5)

In particular if 119894119899119889119890119909(119892) = 1 then the manifold(1198722119899+1

120601 120585 120578 119892 120576) is called a Lorentzian almost paracontactmanifold

If the metric 119892 is positive definite then the manifold(1198722119899+1

120601 120585 120578 119892) is the usual almost paracontact metricmanifold [14]

The fundamental 2-formΦ on119872 is defined by

Φ (119883 119884) = 119892 (119883 120601119884) (6)

Let 1198722119899+1 be an almost paracontact manifold with thestructure (120601 120585 120578) An almost paracomplex structure 119869 on1198722119899+1

timesR1 is defined by

119869 (119883 119891119889

119889119905) = (120601119883 + 119891120585 120578 (119883)

119889

119889119905) (7)

where119883 is tangent to1198722119899+1 119905 is the coordinate onR1 and 119891is a smooth function on1198722119899+1

An almost paracontact structure (120601 120585 120578) is said to benormal if the Nijenhuis tensor 119873

119869of almost paracomplex

structure 119869 defined as

119873119869(119883 119884) = [119869 119869] (119883 119884) = [119869119883 119869119884] + 119869

2

[119883 119884]

minus 119869 [119869119883 119884] minus 119869 [119883 119869119884] (8)

for any vector fields119883119884 isin Γ(119879119872) vanishes

If 119883 and 119884 are vector fields on1198722119899+1 then we have [18ndash20]

119873119869((119883 0) (119884 0))

= (119873120601(119883 119884) minus 2119889120578 (119883 119884) 120585

(L120601119883120578)119884 minus (L

120601119884120578)119883

119889

119889119905)

(9)

119873119869((119883 0) (0

119889

119889119905)) = minus((L

120585120601)119883 ((L

120585120578)119883)

119889

119889119905)

(10)

where 119873120601is Nijenhuis tensor of 120601L

119883is Lie derivative with

respect to a vector field119883 and119873(1) 119873(2) 119873(3) and 119873(4) aredefined as

119873120601(119883 119884)

= [120601 120601] (119883 119884)

= [120601119883 120601119884] + 1206012

[119883 119884] minus 120601 [120601119883 119884] minus 120601 [119883 120601119884]

(11)

119873(1)

(119883 119884) = 119873120601(119883 119884) minus 2119889120578 (119883 119884) 120585 (12)

119873(2)

(119883 119884) = (L120601119883120578)119884 minus (L

120601119884120578)119883 (13)

119873(3)

(119883) = (L120585120601)119883 (14)

119873(4)

(119883) = (L120585120578)119883 (15)

The almost paracontact structure (120601 120585 120578) is normal if andonly if the four tensors119873(1) 119873(2) 119873(3) and 119873(4) vanish

For an almost paracontact structure (120601 120585 120578) vanishing of119873(1) implies the vanishing of119873(2) 119873(3) and119873(4) Moreover

119873(2) vanishes if and only if 120585 is a killing vector fieldAn almost paracontact pseudometric manifold

(1198722119899+1

120601 120585 120578 119892 120576) is called

(i) normal if119873120601minus 2119889120578 otimes 120585 = 0

(ii) paracontact if Φ = 119889120578(iii) 119870-paracontact if119872 is paracontact and 120585 is killing(iv) paracosymplectic if nablaΦ = 0 which implies nabla120578 = 0

where nabla is the Levi-Civita connection on119872(v) almost paracosymplectic if 119889120578 = 0 and 119889Φ = 0(vi) weakly paracosymplectic if119872 is almost paracosym-

plectic and [119877(119883 119884) 120601] = 119877(119883 119884)120601 minus 120601119877(119883 119884) = 0where 119877 is Riemannian curvature tensor

(vii) para-Sasakian if Φ = 119889120578 and119872 is normal(viii) quasi-para-Sasakian if 119889120601 = 0 and119872 is normal

22 Almost Paracomplex Manifolds A (1 1)-type tensor field119869 on 2119898-dimensional smooth manifold 119872 is said to be analmost paracomplex structure if 1198692 = 119868 and (1198722119898 119869) is calledalmost paracomplex manifold

Geometry 3

An almost paracomplex manifold (119872 119869) is such thatthe two eigenbundles 119879

+

119872 and 119879minus

119872 corresponding torespective eigenvalues +1 and minus1 of 119869 have the same rank[21 22]

An almost para-Hermitianmanifold (119872 119869 119892) is a smoothmanifold endowed with an almost paracomplex structure 119869and a pseudo-Riemannian metric 119892 such that

119892 (119869119883 119869119884) = minus119892 (119883 119884) forall119883 119884 isin Γ (119879119872) (16)

Here the metric 119892 is neutral that is 119892 has signature (119898119898)The fundamental 2-form of the almost para-Hermitian

manifold is defined by

119865 (119883 119884) = 119892 (119883 119869119884) (17)

We have the following properties [21 22]

119892 (119869119883 119884) = minus119892 (119883 119869119884) (18)

119865 (119883 119884) = minus119865 (119884119883) (19)

119865 (119869119883 119869119884) = minus119865 (119883 119884) (20)

3119889119865 (119883 119884 119885)

= 119883 (119865 (119884 119885)) minus 119884 (119865 (119883 119885)) + 119885 (119865 (119883 119884))

minus 119865 ([119883 119884] 119885) + 119865 ([119883 119885] 119884) minus 119865 ([119884 119885] 119883)

(21)

(nabla119883119865) (119884 119885) = 119892 (119884 (nabla

119883119869) 119885) = minus119892 (119885 (nabla

119883119869) 119884) (22)

3119889119865 (119883 119884 119885) = (nabla119883119865) (119884 119885) + (nabla

119884119865) (119885119883)

+ (nabla119885119865) (119883 119884)

(23)

the co-differential (120575119865) (119883) =2119898

sum

119894=1

120576119894(nabla119890119894119865) (119890119894 119883) (24)

An almost para-Hermitian manifold is called

(i) para-Hermitian if 119873119869= 0 equivalently (nabla

119869119883119869)119869119884 +

(nabla119883119869)119884 = 0

(ii) para-Kahler if for any 119883 isin Γ(119879119872) nabla119883119869 = 0 that is

nabla119869 = 0(iii) almost para-Kahler if 119889119865 = 0(iv) nearly para-Kahler if (nabla

119883119869)119883 = 0

(v) almost semi-para-Kahler if 120575119865 = 0(vi) semi-para-Kahler if 120575119865 = 0 and119873

119869= 0

23 Pseudo-Riemannian Submersion Let (119872119898

119892) and(119872119899

119892) be two connected pseudo-Riemannian manifolds ofindices 119904 (0 le 119904 le 119898) and 119904 (0 le 119904 le 119899) respectively with119904 ge 119904

A pseudo-Riemannian submersion is a smooth map 119891

119872119898

rarr 119872119899 which is onto and satisfies the following

conditions [2 3 23 24]

(i) The derivative map 119891lowast119901

119879119901119872 rarr 119879

119891(119901)119872 is

surjective at each point 119901 isin 119872

(ii) The fibres 119891minus1(119902) of 119891 over 119902 isin 119872 are eitherpseudo-Riemannian submanifolds of 119872 ofdimension (119898 minus 119899) and index ] or the degeneratesubmanifolds of 119872 of dimension (119898 minus 119899) andindex ] with degenerate metric 119892

|119891minus1(119902)

of type(0 0 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

120583-times minus minus minus minus⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

]-times + + + +⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

(119898minus119899minus120583minus])-times)) where

120583 = dim(V119901capH119901) and ] = 119904 minus 119904 = index of 119892

|119891minus1(119902)

(iii) 119891lowastpreserves the length of horizontal vectors

We denote the vertical and horizontal projections of avector field 119864 on 119872 by 119864V (or by V119864) and 119864

ℎ (or by ℎ119864)respectively A horizontal vector field 119883 on 119872 is said to bebasic if 119883 is 119891-related to a vector field 119883 on119872 Thus everyvector field119883 on119872 has a unique horizontal lift119883 on119872

Lemma 1 (see [1 23]) If 119891 119872 rarr 119872 is a pseudo-Riemannian submersion and119883 119884 are basic vector fields on119872that are 119891-related to the vector fields 119883 119884 on119872 respectivelythen one has the following properties

(i) 119892(119883 119884) = 119892(119883 119884) ∘ 119891(ii) ℎ[119883 119884] is a vector field and ℎ[119883 119884] = [119883 119884] ∘ 119891(iii) ℎ(nabla

119883119884) is a basic vector field 119891-related to nabla

119883119884 where

nabla and nabla are the Levi-Civita connections on119872 and119872respectively

(iv) [119864 119880] isin V for any vector field 119880 isin V and for anyvector field 119864 isin Γ(119879119872)

A pseudo-Riemannian submersion 119891 119872 rarr 119872

determines tensor fieldsT andA of type (1 2) on119872 definedby formulas [1 2 23]

T (119864 119865) = T119864119865 = ℎ (nablaV119864V119865) + V (nablaV119864ℎ119865) (25)

A (119864 119865) = A119864119865 = V (nabla

ℎ119864ℎ119865) + ℎ (nabla

ℎ119864V119865)

for any 119864 119865 isin Γ (119879119872)

(26)

Let 119883 119884 be horizontal vector fields and let 119880 119881 bevertical vector fields on119872 Then one has

T119880119883 = V (nabla

119880119883) T

119880119881 = ℎ (nabla

119880119881) (27)

nabla119880119883 = T

119880119883 + ℎ (nabla

119880119883) (28)

T119883119865 = 0 T

119864119865 = TV119864119865 (29)

nabla119880119881 = T

119880119881 + V (nabla

119880119881) (30)

A119883119884 = V (nabla

119883119884) A

119883119880 = ℎ (nabla

119883119880) (31)

nabla119883119880 = A

119883119880 + V (nabla

119883119880) (32)

A119880119865 = 0 A

119864119865 = A

ℎ119864119865 (33)

nabla119883119884 = A

119883119884 + ℎ (nabla

119883119884) (34)

4 Geometry

ℎ (nabla119880119883) = ℎ (nabla

119883119880) = A

119883119880 (35)

A119883119884 =

1

2V [119883 119884] (36)

A119883119884 = minusA

119884119883 (37)

T119880119881 = T

119881119880 (38)

for all 119864 119865 isin Γ(119879119872)MoreoverT

119880119881 coincideswith second fundamental form

of the submersion of the fibre submanifoldsThe distributionH is completely integrable In view of (37) and (38) A isalternating on the horizontal distribution andT is symmetricon the vertical distribution

3 Paracomplex Paracontact Pseudo-Riemannian Submersions

In this section we introduce the notion of pseudo-Riemannian submersion from almost paracomplex mani-folds onto almost paracontact pseudometric manifolds illus-trate examples and study the transference of structures ontotal manifolds and base manifolds

Definition 2 Let (1198722119898 119869 119892) be an almost para-Hermitianmanifold and let (1198722119899+1 120601 120585 120578 119892) be an almost paracontactpseudometric manifold

A pseudo-Riemannian submersion 119891 119872 rarr 119872 is calledparacomplex paracontact pseudo-Riemannian submersion ifthere exists a 1-form 120578 on119872 such that

119891lowast∘ 119869 = 120601 ∘ 119891

lowast+ 120578 otimes 120585 (39)

Since for each 119901 isin 119872119891lowast119901

is a linear isometry betweenhorizontal spacesH

119901and tangent spaces119879

119891(119901)119872 there exists

an induced almost paracontact structure (120601ℎ

120578ℎ

120585ℎ

119892) on(2119899 + 1)-dimensional horizontal distribution H such that120601ℎ

|

Dℎbehave just like the fundamental collineation of almost

paracomplex structure 119869 on ker 120578ℎ = Dℎ

and 120601ℎ

Dℎ

rarr Dℎ

is an endomorphism such that 120601ℎ

= 119869|ker 120578ℎ

and the rank of

120601ℎ

= 2119899 where dim(Dℎ

) = 2119899It follows that for any 119883ℎ isin D

ℎ

120578ℎ(119883ℎ) = 0 whichimplies that 1198692

|

Dℎ(119883ℎ

) = (120601ℎ

)2

(119883ℎ

) = 119883ℎ for any 119883ℎ isin D

ℎ

andH = Dℎ

oplus 120585ℎ

[18]

Definition 3 (see [25]) A pseudo-Riemannian submersion119891 119872 rarr 119872 is called semi-119869-invariant submersion if thereis a distributionD

1sube ker119891

lowastsuch that

ker119891lowast= D1oplusD2 (40)

119869 (D1) = D

1 119869 (D

2) sube (ker119891

lowast)perp

(41)

whereD2is orthogonal complementary toD

1in ker119891

lowast

Proposition 4 Let119891 1198722119898 rarr 1198722119899+1 be a paracomplex par-

acontact pseudo-Riemannian submersion and let the fibres of119891 be pseudo-Riemannian submanifolds of119872 Then the fibres119891minus1

(119902) 119902 isin 119872 are semi-119869-invariant submanifolds of 119872 ofdimension (2119898 minus 2119899 minus 1)

Proof Let 119880 isinV Then

119891lowast(119869119880) = 120601 (119891

lowast(119880)) + 120578 (119880) 120585

997904rArr 119891lowast119869 (119880) minus 120578 (119880) 120585

ℎ

= 0

(42)

where 119891lowast120585ℎ

= 120585Thus we have

119869 (119880) minus 120578 (119880) 120585ℎ

= 120601 (119880) for some 120601 (119880) isinV (43)

By (19) we get 119892(120585ℎ

119869(120585ℎ

)) = 0 = 119892(120585 119891lowast(119869(120585ℎ

))) = 0As 119892 is nondegenerate on119872 we have

119891lowast(119869 (120585ℎ

)) = 0 that is 119869 (120585ℎ

) isinV (44)

Taking 119880 = 119869120585ℎ

in (43) we obtain

120585ℎ

minus 120578 (119869120585ℎ

) 120585ℎ

= 120601(119869120585ℎ

) (45)

Since fibre 119891minus1(119902) is an odd dimensional submanifold thereexists an associated 1-form 120578

V which is restriction of 120578 on fibresubmanifold 119891minus1(119902) 119902 isin 119872 and a characteristic vector field120585V= 119869120585ℎ

such that 120601(120585V) = 0 So we have 120578V(120585

V) = 1

Let us put ker 120578V = D1andD

2= 120585

V

Then ker119891lowast= D1oplus D2and 119869(D

1) = D

1 119869(D

2) =

119869120585V = 120585

ℎ

sube (ker119891lowast)perp

Hence the fibres 119891minus1(119902) are semi-119869-invariant submani-folds of119872

Corollary 5 Let 119891 1198722119898

rarr 1198722119899+1 be a paracomplex

paracontact pseudo-Riemannian submersion and let the fibresof 119891 be pseudo-Riemannian submanifolds of 119872 Then thefibres 119891

minus1

(119902) are almost paracontact pseudometric mani-folds with almost paracontact pseudo-Riemannian structures(120601

V 120585

V 120578

V 119892

V) 119902 isin 119872 where 120585

V= 119869(120585

ℎ

) 120578V = 120578|V and

119892V= 119892

Proof Since 119891minus1(119902) are semi-119869-invariant submanifolds of119872of odd dimension 2119903 + 1 = 2119898 minus 2119899 minus 1 (39) implies

119869 (119880) = 120601V119880 + 120578

V(119880) 120585ℎ

(46)

for any 119880 isinV

Geometry 5

On operating 119869 on both sides of the above equation weget

119880 = 120601V(120601

V(119880)) + 120578

V(120601

V(119880)) 120585

ℎ

+ 120578V(119880) 120585

V (47)

where 119869(120585ℎ

) = 120585V

Equating horizontal and vertical components we have

119880 = 120601V(120601

V(119880)) + 120578

V(119880) 120585

V 120578

V∘ 120601

V(119880) = 0

997904rArr (120601V)2

(119880) = 119880 minus 120578V(119880) 120585

V 120578

V∘ 120601

V= 0

120601V(120585

V) = 0 120578

V(120585

V) = 1

(48)

Hence (120601V 120585

V 120578

V 119892

V) is almost paracontact pseudometric

structure on the fibre 119891minus1(119902) 119902 isin 119872

Proposition 6 Let 119891 1198722119898

rarr 1198722119899+1 be a paracomplex

paracontact pseudo-Riemannian submersion and let the fibresof 119891 be pseudo-Riemannian submanifolds of 119872 Let 119883 119884 bebasic vector fields 119891-related to 119883 119884 respectively Let 120578 and 120578be 1-forms on the total manifold119872 and the base manifold119872respectively Then one has the following

(i) The characteristic vector field 119869120585ℎ

is a vertical vectorfield

(ii) 119891lowastlowast120578 = 120578ℎ where 119891lowast

lowast120578 is pullback of 120578 through 119891

lowast

(iii) 120578ℎ(119880) = 0 for any vertical vector field 119880(iv) 120578V(119883) = 0 for any horizontal vector field119883

Remark 7 Results (ii) and (iv) are analogue version of results(i) and (iii) of Proposition 4 of [13]

Proof (i) By Corollary 5 (120601V 120585

V 120578

V 119892

V) is almost paracontact

pseudometric structure on 119891minus1(119902) We have

0 = 119892 (120585ℎ

119869120585ℎ

) = 119892(119891lowast(120585ℎ

) 119891lowast(119869120585ℎ

))

= 119892(120585 119891lowast(119869120585ℎ

))

(49)

Now

119891lowast(119869120585ℎ

) = 120601 ∘ 119891lowast120585ℎ

+ 120578 (120585ℎ

) 120585 = 120578 (120585ℎ

) 120585 (50)

so we have

0 = 119892(120585 120578 (120585ℎ

) 120585) = 120578 (120585ℎ

)119892 (120585 120585) = 120578 (120585ℎ

) (51)

Thus 119891lowast(119869120585ℎ

) = 0Hence 119869120585

ℎ

is a vertical vector field(ii) Since 119891 119872

2119898

rarr 1198722119899+1 is smooth submersion

120578ℎ

= 120578|H

is restriction of 120578 on the horizontal distribution

H and 119891lowast119901

H119901rarr 119879119891(119901)

119872 is a linear isometry for any119883119901isinH119901 we get

120578ℎ

119901(119883119901) = 120576119892

119901(120585ℎ

119901 119883119901) = 119892

119891(119901)(119891lowast119901120585ℎ

119901 119891lowast119901119883119901)

= 119892119891(119901)

(120585119891(119901)

119883119891(119901)

) = 120578119891(119901)

(119883119891(119901)

) = 119891lowast

lowast120578119901(119883119901)

(52)

Hence pullback 119891lowastlowast120578 = 120578ℎ

Results (iii) and (iv) immediately follow from the previ-ous results

Example 8 Let (R42 119869 119892) be a paracomplex pseudometric

manifold and let (R31 120601 120585 120578 119892) be an almost paracontact

pseudometric manifoldDefine a submersion 119891 R4

2 (1199091 1199092 1199101 1199102)119905

rarr

R31 (119906 V 119908)119905 by

119891 ((1199091 1199092 1199101 1199102)119905

) 997891997888rarr (1199091+ 1199092+ 31199101+ 21199102

31199091+ 21199092+ 1199101+ 1199102

51199091+ 31199092+ 51199101+ 31199102)119905

(53)

Then the kernel of 119891lowastis

V = ker119891lowast= Span119881

1=

120597

1205971199091

minus 2120597

1205971199092

minus120597

1205971199101

+ 2120597

1205971199102

(54)

which is the vertical distribution admitting one lightlikevector field that is fibre is degenerate submanifold of R4

2

The horizontal distribution is

H = (ker119891lowast)perp

= Span1198831=

120597

1205971199091

minus120597

1205971199101

1198832=

120597

1205971199092

+ 2120597

1205971199101

1198833= 2

120597

1205971199101

+120597

1205971199102

(55)

For any real 119896 the horizontal characteristic vector field 120585ℎ

isgiven by

120585ℎ

= 119896120597

1205971199091

minus (2119896 minus1

3)

120597

1205971199092

minus (119896 minus 1)120597

1205971199101

+ (2119896 minus5

3)

120597

1205971199102

(56)

which is 119891-related to the characteristic vector field 120585 = 120597120597119908Moreover there exists one form 120578 = 5119889119909

1+3119889119909

2+51198891199101+

31198891199102on (R4

2 119869 119892) such that the submersion satisfies (39)

Example 9 Let (R63 119869 119892) be an almost paracomplex pseudo-

Riemannian manifold and let (R31 120601 120585 120578 119892) be an almost

6 Geometry

paracontact pseudo-Riemannian manifold Consider a sub-mersion 119891 R6

3 (1199091 1199092 1199093 1199101 1199102 1199103)119905

rarr R31 (119906 V 119908)119905

defined by

119891 ((1199091 1199092 1199093 1199101 1199102 1199103)119905

)

997891997888rarr (1199091+ 1199092

radic2

1199101+ 1199102

radic2

1199102+ 1199103

radic2

)

119905

(57)

Then there exists one form 120578 = (1198891199092+ 1198891199093)radic2 on (R6

3 119869 119892)

such that (39) is satisfied The kernel of 119891lowastis

V = ker119891lowast

= Span1198811=

120597

1205971199091

minus120597

1205971199092

1198812=

120597

1205971199101

minus120597

1205971199102

+120597

1205971199103

1198813=

120597

1205971199093

(58)

which is vertical distribution admitting non-lightlike vectorfields that is the fibre is nondegenerate submanifold of(R63 119869 119892)The horizontal distribution is

H = Span1198831=

120597

1205971199091

+120597

1205971199092

1198832= minus

120597

1205971199101

+120597

1205971199103

1198833=

120597

1205971199101

+120597

1205971199102

(59)

Example 10 Let (R42 119869 119892) be a paracomplex pseudometric

manifold and let (R31 120601 120585 120578 119892) be an almost paracontact

pseudometric manifoldConsider a submersion 119891 R4

2 (1199091 1199092 1199101 1199102)119905

rarr R31

(119906 V 119908)119905 defined by

119891 ((1199091 1199092 1199101 1199102)119905

) 997891997888rarr (1199091 1199101 1199102)119905

(60)

Then the kernel of 119891lowastis

V = ker119891lowast= Span119881

1=

120597

1205971199092

(61)

which is the vertical distribution and the restriction of 119892 tothe fibres of 119891 is nondegenerate

The horizontal distribution is

H = (ker119891lowast)perp

= Span119883 =120597

1205971199091

119884 =120597

1205971199101

120585ℎ

=120597

1205971199102

(62)

The characteristic vector field 120585 = 120597120597119908 on R31has unique

horizontal lift 120585ℎ

which is the characteristic vector field onhorizontal distributionH of R4

2

We also have

119892 (119883119883) = 119892 (119891lowast119883119891lowast119883) = minus1

119892 (119884 119884) = 119892 (119891lowast119884 119891lowast119884) = 1

119892 (120585ℎ

120585ℎ

) = 119892(119891lowast120585ℎ

119891lowast120585ℎ

) = 119892 (120585 120585) = 1

(63)

Thus the smooth map 119891 is a pseudo-Riemannian submer-sion

Moreover we obtain that there exists a 1-form 120578 = 1198891199092on

R42such that 120578(119869120585

ℎ

) = 1 120578(120585ℎ

) = 0 and the map 119891 satisfies

119891lowast119869119883 = 120601119891

lowast119883 + 120578 (119883) 120585

119891lowast119869119884 = 120601119891

lowast119884 + 120578 (119884) 120585

119891lowast119869120585ℎ

= 120601119891lowast120585ℎ

+ 120578 (120585ℎ

) 120585

(64)

Hence the map 119891 is a paracomplex paracontact pseudo-Ri-emannian submersion from R4

2on to R3

1

Proposition 11 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion and let the fibresof 119891 be pseudo-Riemannian submanifolds of 119872 Let 119883 119884 bebasic vector fields 119891-related to119883 119884 respectively Then 119869(119883) minus120576119892(119883 120585

ℎ

)120585ℎ

is 119891-related to 120601119883

Proof Since119883 is 119891-related to vector field119883 on119872 we have

120578 (119883) = 120578V+ 120578ℎ

(119883) = 0 + 120578ℎ

(119883) = 120576119892 (119883 120585ℎ

)

997904rArr 119891lowast(119869119883) = 120601119883 + 120578

ℎ

(119883) 120585

997904rArr 119891lowast119869119883 minus 120576119892 (119883 120585

ℎ

) 120585ℎ

= 120601119883

(65)

Hence 119869(119883) minus 120576119892(119883 120585ℎ

)120585ℎ

is 119891-related to 120601119883

Proposition 12 Let 119891 119872 rarr 119872 be a paracomplex para-contact pseudo-Riemannian submersion and let the fibres of 119891be pseudo-Riemannian submanifolds of 119872 Let V and H bethe vertical and horizontal distributions respectively If 120585

ℎ

isthe basic characteristic vector field of horizontal distribution119891-related to the characteristic vector field 120585 of base manifoldthen

(i) 119869V sub DVoplus 119869120585ℎ

oplus 120585ℎ

(ii) 119869H sub Dℎ

oplus 120585ℎ

oplus 119869120585ℎ

Proof (i) Let 119880 isin V Then 119880 = 119886119880|DV + 119887119869120585

ℎ

for 119886 119887 isin

119862infin

(119872) as 119869120585ℎ

= 120585Vis characteristic vector field on odd

Geometry 7

dimensional fibre submanifold 119891minus1(119902) of119872 119902 isin 119872 We get

119869119880 = 119886119869119880|DV + 119887119869

2

120585ℎ

= 119886119869119880|DV + 119887120585

ℎ

isinV oplus 120585ℎ

997904rArr 119869V subV oplus 120585ℎ

(66)

Again let 119881 isinV oplus 120585ℎ

Then 119881 = 119886119881|DV + 119887119869120585

ℎ

+ 119888120585ℎ

where

120578V(119881|DV ) = 0 D

V= ker 120578V 119886119881

|DV + 119887119869120585

ℎ

isin V and 119886 119887 119888 isin119862infin

(119872) We have

119869119881 = 119886119869119881|DV + 119887120585

ℎ

+ 119888119869120585ℎ

= (119886119869119881|DV + 119888119869120585

ℎ

)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

isinV

+ 119887120585ℎ

⏟⏟⏟⏟⏟⏟⏟

isin120585

ℎ

isinV oplus 120585ℎ

(67)

Now by (39) we get

119891lowast119869119881 = 120601 (119891

lowast119881) + 120578 (119891

lowast(119881)) 120585

= 119888 120601 (119891lowast120585ℎ

) + 120578 (120585) 120585

= 119888120585 isin 120585 sube 119869V

(68)

We get 119869119881 notinVHence 119869V subV oplus 120585

ℎ

that is 119869V sub DVoplus 119869120585ℎ

oplus 120585ℎ

(ii) Let 119883 = 119886119883

|

Dℎ+ 119887120585ℎ

isin H where H = Dℎ

oplus

120585ℎ

ker 120578ℎ = Dℎ

and 119886 119887 isin 119862infin(119872) Then

119869119883 = 119886119869119883|

Dℎ+ 119887119869120585ℎ

isinH oplus 119869120585ℎ

(69)

which implies that 119869H subH oplus 119869120585ℎ

Again let119884 isinHoplus119869120585

ℎ

Then119884 = 119886119884|

Dℎ+119887120585ℎ

+119888119869120585ℎ

notinHfor 119886 119887 119888 isin 119862infin(119872) We have

119869 119884 = 119886119869119884|

Dℎ+ 119887119869120585ℎ

+ 1198881198692

120585ℎ

= 119886119869119884|

Dℎ+ 119887119869120585ℎ

+ 119888120585ℎ

= 119885 + 119887119869120585ℎ

isinH oplus 119869120585ℎ

for some 119885 = 119886119869119884|

Dℎ+ 119888120585ℎ

isinH

(70)

We obtain 119869 119884 notinHHence 119869H sub H oplus 119869(120585

ℎ

) that is 119869H sub Dℎ

oplus 120585ℎ

oplus

119869120585ℎ

Example 13 Let (R63 119869 119892) be an almost paracomplex pseudo-

Riemannian manifold and let (R31 120601 120585 120578 119892) be an almost

paracontact pseudo-Riemannian manifold Consider a sub-mersion 119891 R6

3 (1199091 1199092 1199093 1199101 1199102 1199103)119905

rarr R31 (119906 V 119908)119905

defined by

119891 ((1199091 1199092 1199093 1199101 1199102 1199103)119905

) 997891997888rarr (1199091+ 1199092

radic2

1199101+ 1199102

radic2

1199103)

119905

(71)

Then the kernel of 119891lowastis

V = ker119891lowast

= Span1198811=

120597

1205971199091

minus120597

1205971199092

1198812=

120597

1205971199101

minus120597

1205971199102

120585V=

120597

1205971199093

(72)

which is the vertical distribution and the restriction of 119892 tothe fibres of 119891 is nondegenerate

The horizontal distribution is

H = (ker119891lowast)perp

= Span1198831=

120597

1205971199091

+120597

1205971199092

1198832=

120597

1205971199101

+120597

1205971199102

120585ℎ

=120597

1205971199103

(73)

The characteristic vector field 120585 = 120597120597119908 on R31has unique

horizontal lift 120585ℎ

which is the characteristic vector field onthe horizontal distributionH of R6

3

We also have

119892 (1198831 1198831) = 119892 (119891

lowast1198831 119891lowast1198831) = minus2

119892 (1198832 1198832) = 119892 (119891

lowast1198832 119891lowast1198832) = 2

119892 (120585ℎ

120585ℎ

) = 119892(119891lowast120585ℎ

119891lowast120585ℎ

) = 119892 (120585 120585) = 1

(74)

Thus the smooth map 119891 is a pseudo-Riemannian submer-sion

Also we obtain that there exists a 1-form 120578 = 1198891199093on R63

such that 120578(119869120585ℎ

) = 1 120578(120585ℎ

) = 0 and the map 119891 satisfies

119891lowast1198691198831= 120601119891lowast1198831+ 120578 (119883

1) 120585

119891lowast1198691198832= 120601119891lowast1198832+ 120578 (119883

2) 120585

119891lowast119869120585ℎ

= 120601119891lowast120585ℎ

+ 120578 (120585ℎ

) 120585

(75)

Hence the map 119891 is a paracomplex paracontact pseudo-Riemannian submersion from R6

3onto R3

1

Moreover we observe that for this submersion119891 we have

119869V subV oplus 120585ℎ

119869H subH oplus 119869120585ℎ

(76)

which verifies Proposition 12

8 Geometry

Proposition 14 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion and let the fibresof 119891 be pseudo-Riemannian submanifolds of 119872 Let 119883 119884 bebasic vector fields 119891-related to 119883 119884 respectively Let 119865 and Φbe the second fundamental forms and let nabla and nabla be the Levi-Civita connection on the total manifold119872 and base manifold119872 respectively Then one has

(i) 119891lowast((nabla119883119869)119884) = (nabla

119883120601)119884 + 120576119892(119884 nabla

119883120585)120585 + 120578(119884)nabla

119883120585

(ii) 119865 = 119891lowastlowastΦ + 120576120578 otimes 120578

(iii) 119891lowast((nabla119883119865)(119884 119885)) = (nabla

119883Φ)(119884 119885) + 120578(119884)119892(119885 nabla

119883120585) +

120578(119885)119892(119884 nabla119883120585)

Proof (i) In view of Definition 2 and Proposition 11 we have

119891lowast((nabla119883119869) 119884) = 119891

lowast(nabla119883(119869 119884) minus 119869 (nabla

119883119884))

= nabla119883(119891lowast(119869 119884)) minus 119891

lowast(119869 (nabla119883119884))

= nabla119883(120601119884) + nabla

119883(120578 (119884) 120585) minus 120601 (nabla

119883119884)

minus 120578 (nabla119883119884) 120585

= (nabla119883120601)119884 + nabla

119883(120576119892 (119884 120585) 120585) minus 120578 (nabla

119883119884) 120585

= (nabla119883120601)119884 + 120576119892 (nabla

119883119884 120585) 120585 + 120576119892 (119884 nabla

119883120585) 120585

+ 120576119892 (119884 120585) nabla119883120585 minus 120576119892 (nabla

119883119884 120585) 120585

= (nabla119883120601)119884 + 120576119892 (119884 nabla

119883120585) 120585 + 120578 (119884) nabla

119883120585

(77)

(ii) Since119891lowastlowastΦ is pullback ofΦ through the linearmap119891

lowast

we get

119891lowast

lowastΦ(119883 119884) = Φ (119883 119884) ∘ 119891 = 119892 (119883 120601119884) ∘ 119891

= 119892 (119883 119869 119884) minus 120576120578 (119884) 120578 (119883)

= 119865 (119883 119884) minus 120576120578 (119883) 120578 (119884)

(78)

which implies 119865 = 119891lowastlowastΦ + 120576120578 otimes 120578

(iii) By (23) we have

119891lowast((nabla119883119865) (119884 119885)) = 119892 (119891

lowast(119884) 119891

lowast((nabla119883119869)119885)) (79)

Now using (i) in the above equation we get (iii)

Theorem 15 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion and let the fibresof 119891 be pseudo-Riemannian submanifolds of 119872 Let 119883 119884 bebasic vector fields 119891-related to 119883 119884 respectively If the totalspace is para-Hermitian manifold then the almost paracontactstructure of base space is normal

Moreover if the almost paracontact structure of base spaceis normal then the Nijenhuis tensor of total space is vertical

Proof The Nijenhuis tensors 119873119869and 119873

120601of almost para-

complex structure 119869 and almost paracontact structure 120601 arerespectively defined by (8) and (11)

Using Definition 2 and properties of Sections 21 and 22we get the following identity

119891lowast(119873119869(119883 119884)) = 119873

(1)

(119883 119884) + 2119889120578 (120601119883 119884) 120585

minus 2119889120578 (120601119884119883) 120585

+ 2120578 (119883) 119889120578 (120585 119884) 120585 minus 2120578 (119884) 119889120578 (120585 119883) 120585

minus 120578 (119884)119873(3)

(119883) + 120578 (119883)119873(3)

(119884)

(80)

Using (12) (13) (14) and (15) (80) reduces to

119891lowast(119873119869(119883 119884)) = 119873

(1)

(119883 119884) + 119873(2)

(119883 119884) 120585

+ 120578 (119883)119873(4)

(119884) 120585

minus 120578 (119884)119873(4)

(119883) 120585 minus 120578 (119884)119873(3)

(119883)

+ 120578 (119883)119873(3)

(119884)

(81)

Since 119873119869(119883 119884) = 0 it follows from (81) that

tensors 119873(1) 119873(2) 119873(3) and 119873(4) vanish togetherHence the almost paracontact structure of base space is

normalConversely let the almost paracontact structure of the

base space be normalThen (81) implies that 119891

lowast(119873119869(119883 119884)) = 0

Hence119873119869(119883 119884) is vertical

Corollary 16 Let119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion and let the fibres of 119891 bepseudo-Riemannian submanifolds of 119872 Let 119883 119884 be basicvector fields 119891-related to 119883 119884 respectively Let the total spacebe para-Hermitian manifold and119873(1) vanishes Then the basespace is paracontact pseudometric manifold if and only if 120585 iskilling

Proof Let the total space be para-Hermitian and 119873(1) van-ishes Then from (80) we have

0 = 2119889120578 (120601119883 119884) 120585 minus 2119889120578 (120601119884119883) 120585 + 2120578 (119883) 119889120578 (120585 119884) 120585

minus 2120578 (119884) 119889120578 (120585 119883) 120585 minus 120578 (119884) (L120585120601)119883 + 120578 (119883) (L

120585120601)119884

(82)

If 120585 is killing then we have L120585120601 = 0 It immediately follows

from (82) that

119889120578 (120601119883 119884) minus 120578 (120601119884119883) + 120578 (119883) 119889120578 (120585 119884)

minus 120578 (119884) 119889120578 (120585 119883) = 0

(83)

In view of (6) and (7) the above equation gives 119889120578 = ΦConversely let the base space be paracontact Then 119889120578 =

ΦUsing (6) (7) and (82) we getL

120585120601 = 0

Hence the characteristic vector field 120585 is killing

Geometry 9

Theorem 17 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion and let the fibres of 119891 bepseudo-Riemannian submanifolds of 119872 Let 119883 119884 be basicvector fields 119891-related to 119883 119884 respectively If the total spaceis para-Kahler then the base space is paracosymplectic Theconverse is true if nabla

119883119869 is vertical

Proof We have for any 119883119884 isin Γ(119879119872) (nabla119883120601)119884 = 0 which

gives 119892(119885 (nabla119883120601)119884) = 0 for any 119885 isin Γ(119879119872)

From Proposition 14 we have

119892 (119885 (nabla119883120601)119884) + 120576120578 (119884) (nabla

119883120578)119885 + 120576120578 (119885) (nabla

119883120578) 119884 ∘ 119891

= 119892 (119885 (nabla119883119869) 119884)

(84)

Let nabla 119869 = 0 that is the total space is para-KahlerThen from(84) we obtain nabla120601 = 0 and nabla120578 = 0 Hence the base space isparacosymplectic

Again let (nabla119883120601)119884 = 0 and nabla120578 = 0 Then 119892(119885 (nabla

119883119869)119884) =

0 which implies that (nabla119883119869)119884 is a vertical vector field

Theorem 18 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion and let the fibres of 119891 bepseudo-Riemannian submanifolds of 119872 Let 119883 119884 and 119885 bebasic vector fields 119891-related to 119883 119884 and 119885 respectively If120578(119883)119885(120578(119884))+120578(119884)119883(120578(119885))minus120578(119883)119884(120578(119885)) = 0 then the totalspace is almost para-Kahler if and only if the base space119872 isan almost paracosymplectic manifold

Proof We have the following equation

3119889119865 (119883 119884 119885)

= 3 (119891lowast

lowast119889Φ) (119883 119884 119885) + 2120576120578 (119885) 119889120578 (119883 119884)

minus 2120576120578 (119884) 119889120578 (119883 119885)

+ 2120576120578 (119883) 119889120578 (119884 119885) + 2120576120578 (119883)119885 (120578 (119884))

+ 2120576120578 (119884)119883 (120578 (119885))

minus 2120576120578 (119883)119884 (120578 (119885))

(85)

If 119889120578 = 0 119889Φ = 0 and 120578(119883)119885(120578(119884)) + 120578(119884)119883(120578(119885)) minus

120578(119883)119884(120578(119885)) = 0 then from (85) we have 119889119865 = 0 Hencethe total space is almost para-Kahler

Conversely let 119889119865 = 0 and 120578(119883)119885(120578(119884))+120578(119884)119883(120578(119885))minus120578(119883)119884(120578(119885)) = 0

By using the above equation in (85) we have 119889120578 = 0 and119889Φ = 0

Hence the base space is almost paracosymplectic

Nowwe investigate the properties of fundamental tensorsT andA of a pseudo-Riemannian submersion

Lemma 19 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometric manifold119872 and

let the fibres of 119891 be pseudo-Riemannian submanifolds of 119872Then for any horizontal vector fields119883 119884 and for any verticalvector fields 119880 119881 on119872 one has

(i) A119883(119869 119884) = 119869(A

119883119884)

(ii) A119869119883(119884) = 119869(A

119883119884)

(iii) T119880(119869 119881) = 119869(T

119880119881)

(iv) T119869119880119881 = 119869(T

119880119881)

Proof The proof follows using similar steps as in Lemmas 3and 4 of [13] so we omit it

Lemma 20 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometric manifold119872 andlet the fibres of 119891 be pseudo-Riemannian submanifolds of 119872Then for any vector fields 119864 119865 on119872 one has

(i) A119864(119869119865) = 119869(A

119864119865)

(ii) T119864(119869119865) = 119869(T

119864119865)

Proof The proof follows from (37) and (38)

Theorem 21 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometric manifold119872 andlet the fibres of 119891 be pseudo-Riemannian submanifolds of 119872Then the horizontal distribution is integrable

Proof For any vertical vector field 119880 we have

119892 (119869 (A119883119884) 119880) = 119892 (A

119883119869 119884119880)

= minus119892 (119869119884A119883119880)

= minus119892 (119869119884 ℎ (nabla119880119883))

= 119892 (119884 ℎ (119869 (nabla119880119883)))

= 119892 (119884 ℎ (minusnabla119880119869)119883

+nabla119880(119869119883))

= 119892 (119884 ℎ nabla119880(119869119883)) = 119892 (119884A

119869119883119880)

= minus119892 (A119869119883119884119880) = minus119892 (119869 (A

119883119884) 119880)

(86)

Thus 119892(119869(A119883119884) 119880) = 0 which is true for all119883 and 119884

SoA119883119884 = 0

Hence the horizontal distribution is integrable

Theorem 22 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometric manifold119872 andlet the fibres of 119891 be pseudo-Riemannian submanifolds of 119872Then the submersion is an affine map onH

10 Geometry

Proof The second fundamental form of 119891 is defined by

(nabla119891lowast) (119864 119865) = (nabla

119891

119864119891lowast(119865)) ∘ 119891 minus 119891

lowast(nabla119864119865) (87)

where 119864 119865 isin Γ(119879119872) and nabla119891 is pullback connection of Levi-Civita connection nabla on119872 with respect to 119891

We have for any119883119884 isinH

(nabla119891lowast) (119883 119884) = (nabla

119891

119883

119891lowast(119884)) ∘ 119891 minus 119891

lowast(nabla119883119884) (88)

By using Lemma 1 we have 119891lowast(ℎ(nabla119883119884)) = (nabla

119883119884) ∘ 119891 which

implies nabla119891lowast= 0

Hence the submersion 119891 is an affine map onH

Theorem 23 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Hermitian man-ifold119872 onto an almost paracontact pseudometric manifold119872and let the fibres of 119891 be pseudo-Riemannian submanifolds of119872 Then the submersion is an affine map on V if and only ifthe fibres of 119891 are totally geodesic

Proof We have for any 119880119881 isinV

(nabla119891lowast) (119880 119881) = minus119891

lowast(ℎ (nabla119880119881)) (89)

which in view of (27) gives

(nabla119891lowast) (119880 119881) = minus119891

lowast(T119880119881) (90)

Let the fibres of 119891 be totally geodesic Then T = 0Consequently from the above equation we have nabla119891

lowast= 0

Thus the map 119891 is affine onVConversely let the submersion 119891 be an affine map onV

Then nabla119891lowast= 0 which impliesT = 0

Hence the fibres of 119891 are totally geodesic

Theorem 24 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Hermitian man-ifold119872 onto an almost paracontact pseudometric manifold119872and let the fibres of 119891 be pseudo-Riemannian submanifoldsof 119872 Then the submersion is an affine map if and only ifℎ(nabla119864ℎ119865)+A

ℎ119864V119865+TV119864V119865 is 119891-related to nabla

119883119884 for any 119864 119865 isin

Γ(119879119872)

Proof For any 119864 119865 isin Γ(119879119872) with 119891lowastℎ119864 = 119883 ∘ 119891 and 119891

lowastV119865 =

119884 ∘ 119891 we have

(nabla119891lowast) (119864 119865) = (nabla

119891lowastℎ119864(119891lowastℎ119865)) ∘ 119891 minus 119891

lowast(ℎ (nabla119864119865))

= (nabla119883119884) ∘ 119891 minus 119891

lowast(ℎ (nablaℎ119864ℎ119865 + nabla

ℎ119864V119865

+nablaV119864ℎ119865 + nablaV119864ℎ119865))

(91)

By using (27) and (31) in the above equation we have

(nabla119891lowast) (119864 119865) = (nabla

119883119884) ∘ 119891 minus 119891

lowast(ℎ (nabla119864ℎ119865) +A

ℎ119864V119865

+TV119864V119865) (92)

Let the submersion map be affine Then for any 119864 119865 isin

Γ(119879119872) (nabla119891lowast)(119864 119865) = 0 Equation (92) implies (nabla

119883119884) ∘ 119891 =

119891lowast(ℎ(nabla119864ℎ119865) +A

ℎ119864V119865 +TV119864ℎ119865)

Conversely let ℎ(nabla119864119865) + A

ℎ119864V119865 + TV119864ℎ119865 be 119891-related

to nabla119883119884 for any 119864 119865 isin Γ(119879119872) Then from (92) we have

(nabla119891lowast)(119864 119865) = 0Hence the submersion map 119891 is affine

4 Curvature Properties

In this section the paraholomorphic bisectional curvaturesand paraholomorphic sectional curvatures of total mani-fold base manifold and fibres of paracomplex paracontactpseudo-Riemannian submersion and their curvature proper-ties are studied

Let 119891 119872 rarr 119872 be a paracomplex paracontact pseudo-Riemannian submersion from an almost para-Hermitianmanifold (119872 119869 119892) onto an almost paracontact pseudometricmanifold (119872 120601 120585 120578 119892)

Suppose that the vector fields 119864 119865 span the 2-dimensional plane at point 119901 of 119872 and let R be theRiemannian curvature tensor of 119872 The paraholomorphicbisectional curvature 119861(119864 119865) of 119872 for any pair of nonzeronon-lightlike vector fields 119864 119865 on 119872 is defined by theformula

119861 (119864 119865) =

R (119864 119869119864 119865 119869119865)

119892 (119864 119864) 119892 (119865 119865) (93)

For a nonzero non-lightlike vector field 119864 the vector field119869119864 is also non-lightlike and 119864 119869119864 span the 2-dimensionalplane Then the paraholomorphic sectional curvature 119867(119864)is defined as

119867(119864) = 119861 (119864 119864) =

R (119864 119869119864 119864 119869119864)

119892 (119864 119864) 119892 (119864 119864) (94)

The curvature properties of Riemannian submersion andsemi-Riemannian submersion have been extensively studiedin the work of OrsquoNeill [1] and Gray [3]

Let 119861ℎ and 119861V be the paraholomorphic bisectional cur-vatures of horizontal and vertical spaces respectively Let119867ℎ and 119867V be the paraholomorphic sectional curvatures of

horizontal and vertical spaces respectively Let 119861 and 119867 bethe paraholomorphic bisectional and sectional curvatures ofthe base manifold respectively

Proposition 25 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion from an almostpara-Hermitian manifold onto an almost paracontact pseudo-metric manifold and let the fibres of 119891 be pseudo-Riemanniansubmanifolds of 119872 Let 119880 119881 be non-lightlike unit verticalvector fields and let119883 119884 be non-lightlike unit horizontal vectorfields on119872 Then one has

119861 (119880119881) = 119861V(119880 119881) + 119892 (T

119880(119869119881) T

119869119880119881)

minus 119892 (T119869119880(119869119881) T

119880119881)

(95)

Geometry 11

119861 (119883119880) = 119892 ((nabla119880A)119883

119869119883 119869119880) minus 119892 (A119883119869119880A

119869119883119880)

+ 119892 (A119883119880A119869119883119869119880) minus 119892 ((nabla

119869119880A)119883

119869119883119880)

+ 119892 (T119869119880119883T119880(119869119883))

minus 119892 (T119880119883T119869119880(119869119883))

(96)

119861 (119883 119884) = 119861ℎ

(119883 119884) minus 2119892 (A119883(119869119883) A

119884(119869 119884))

+ 119892 (A119869119883119884A119883(119869 119884))

minus 119892 (A119883119884A119869119883(119869 119884))

(97)

Proof Using Definitions (93) and (94) of paraholomorphicsectional curvature and fundamental equations of submer-sion obtained by OrsquoNeill [1] we have (95) (96) and (97)

Corollary 26 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion from an almostpara-Hermitian manifold onto an almost paracontact pseudo-metric manifold If the fibres of 119891 are totally geodesic pseudo-Riemannian submanifolds of119872 then for any non-lightlike unitvertical vector fields 119880 and 119881 one has

119861 (119880119881) = 119861V(119880 119881) (98)

Corollary 27 Let 119891 119872 rarr 119872 be a paracomplex para-contact pseudo-Riemannian submersion from an almost para-Hermitian manifold onto an almost paracontact pseudometricmanifold and let the fibres of 119891 be totally geodesic pseudo-Riemannian submanifolds of119872 If the horizontal distributionis integrable then for any non-lightlike unit horizontal vectorfields119883 and 119884 one has

119861 (119883 119884) = 119861ℎ

(119883 119884) (99)

Proposition 28 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion from an almostpara-Hermitian manifold onto an almost paracontact pseudo-metric manifold and let the fibres of 119891 be pseudo-Riemanniansubmanifolds of119872 Let 119880 and 119883 be non-lightlike unit verticalvector field and non-lightlike unit horizontal vector fieldrespectively Then one has

119867(119880) = 119867V(119880) +

10038171003817100381710038171003817T119880(119869119880)

10038171003817100381710038171003817

2

minus 119892 (T119869119880(119869119880) T

119880119880)

(100)

119867(119883) = 119867 (119883) ∘ 119891 minus 310038171003817100381710038171003817A119883(119869119883)

10038171003817100381710038171003817

2

(101)

Proof The proof is straightforward If we take 119880 = 119881 in (95)and119883 = 119884 in (97) we have (98) and (99)

Corollary 29 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion from an almostpara-Hermitian manifold onto an almost paracontact pseudo-metric manifold If the fibres of 119891 are totally geodesic pseudo-Riemannian submanifolds of 119872 then the total manifold andfibres of119891have the same paraholomorphic sectional curvatures

Proof Since the fibres are totally geodesic T = 0 conse-quently we have

119867(119880) = 119867V(119880) (102)

Corollary 30 Let 119891 119872 rarr 119872 be a paracomplex para-contact pseudo-Riemannian submersion from an almost para-Hermitian manifold onto an almost paracontact pseudometricmanifold and let the fibres of 119891 be totally geodesic pseudo-Riemannian submanifolds of119872 If the horizontal distributionis integrable then the base manifold and horizontal distribu-tion have the same paraholomorphic sectional curvatures

Proof Since the horizontal distribution is integrableA = 0consequently we have

119867(119883) = 119867 (119883) ∘ 119891 (103)

Theorem 31 Let 119891 119872119898

rarr 119872119899 be a paracomplex

paracontact pseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometricmanifold 119872 and let the fibres of 119891 be pseudo-Riemanniansubmanifolds of 119872 If 119880 119881 are the non-lightlike unit verticalvector fields and 119883 119884 are the non-lightlike unit horizontalvector fields then one has

119861 (119880119881) = 119861V(119880 119881) (104)

119861 (119883119880) = minus210038171003817100381710038171003817T11988011988310038171003817100381710038171003817

2

(105)

119861 (119883 119884) = 119861 (119883 119884) ∘ 119891 (106)

Proof Using results of Lemma 19 in (95) we have

119861 (119880119881) = 119861V(119880 119881) minus 119892 (119869 (T

119880119881) 119869 (T

119880119881))

minus 119892 (1198692

(T119880119881) T

119880119881)

= 119861 (119880119881) + 119892 (T119880119881T119880119881) minus 119892 (T

119880119881T119880119881)

= 119861V(119880 119881)

(107)

Applying results of Lemma 19 in (96) we have

119861 (119883119880) = 119892 ((nabla119880A)119883

(119869119883) 119869119880) minus 119892 ((nabla119869119880A)119883

(119869119883) 119880)

+ 21003817100381710038171003817A119883119880

1003817100381710038171003817

2

minus 210038171003817100381710038171003817T11988011988310038171003817100381710038171003817

2

(108)

Since byTheorem 21 the horizontal distribution is integrablewe haveA = 0 which implies

119861 (119883119880) = minus210038171003817100381710038171003817T11988011988310038171003817100381710038171003817

2

(109)

In view ofA = 0 (104) follows from (97)

12 Geometry

Theorem 32 Let 119891 119872119898

rarr 119872119899 be a paracomplex

paracontact pseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometricmanifold 119872 and let the fibres of 119891 be pseudo-Riemanniansubmanifolds of119872 If 119880 119883 are non-lightlike unit vertical andnon-lightlike unit horizontal vector fields respectively then onehas

119867(119880) = 119867V(119880) minus 2

1003817100381710038171003817T1198801198801003817100381710038171003817

2

(110)

119867(119883) = 119867 (119883) ∘ 119891 (111)

Proof Since 119891 is the paracomplex paracontact pseudo-Riemannian submersion from a para-Kahler manifold 119872

onto an almost paracontact pseudometric manifold 119872 by(16) and equations of Lemma 19 we have

119892 (T119869119880(119869119880) T

119880119880) = 119892 (119869

2

(T119880119880) T

119880119880) =

1003817100381710038171003817T1198801198801003817100381710038171003817

2

119892 (T119880(119869119880) T

119880(119869119880)) = minus119892 (T

119880119880T119880119880) = minus

1003817100381710038171003817T1198801198801003817100381710038171003817

2

(112)

and by using the above results in (100) we have

119867(119880) = 119867V(119880) minus

1003817100381710038171003817T1198801198801003817100381710038171003817

2

minus1003817100381710038171003817T119880119880

1003817100381710038171003817

2

= 119867V(119880) minus 2

1003817100381710038171003817T1198801198801003817100381710038171003817

2

(113)

Again since horizontal distribution is integrable we haveA = 0 and putting it in (101) we obtain (111)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Uma Shankar Verma is thankful to University Grant Com-mission New Delhi India for financial support

References

[1] B OrsquoNeill ldquoThe fundamental equations of a submersionrdquo TheMichigan Mathematical Journal vol 13 pp 459ndash469 1966

[2] B OrsquoNeill Semi-Riemannian Geometry with Applications toRelativity vol 103 of Pure and Applied Mathematics AcademicPress New York NY USA 1983

[3] A Gray ldquoPseudo-Riemannian almost product manifolds andsubmersionsrdquo vol 16 pp 715ndash737 1967

[4] J P Bourguignon and H B Lawson A Mathematicians Visitto Kaluza-Klein Theory Rendiconti del Seminario Matematico1989

[5] S Ianus and M Visinescu ldquoSpace-time compactification andRiemannian submersionsrdquo in The Mathematical Heritage GRassias and C F Gauss Eds pp 358ndash371 World ScientificRiver Edge NJ USA 1991

[6] J P Bourguignon and H B Lawson Jr ldquoStability and isolationphenomena for Yang-Mills fieldsrdquo Communications in Mathe-matical Physics vol 79 no 2 pp 189ndash230 1981

[7] B Watson ldquoG 1198661015840

-Riemannian submersions and nonlineargauge field equations of general relativityrdquo in Global AnalysismdashAnalysis on Manifolds T Rassias and M Morse Eds vol 57 ofTeubner-Texte zur Mathematik pp 324ndash349 Teubner LeipzigGermany 1983

[8] C Altafini ldquoRedundant robotic chains on Riemannian submer-sionsrdquo IEEE Transactions on Robotics and Automation vol 20no 2 pp 335ndash340 2004

[9] M T Mustafa ldquoApplications of harmonic morphisms to grav-ityrdquo Journal of Mathematical Physics vol 41 no 10 pp 6918ndash6929 2000

[10] B Watson ldquoAlmost Hermitian submersionsrdquo Journal of Differ-ential Geometry vol 11 no 1 pp 147ndash165 1976

[11] D Chinea ldquoAlmost contact metric submersionsrdquo Rendiconti delCircolo Matematico di Palermo II vol 34 no 1 pp 319ndash3301984

[12] D Chinea ldquoTransference of structures on almost complexcontact metric submersionsrdquo Houston Journal of Mathematicsvol 14 no 1 pp 9ndash22 1988

[13] Y Gunduzalp and B Sahin ldquoPara-contact para-complexpseudo-Riemannian submersionsrdquo Bulletin of the MalaysianMathematical Sciences Society

[14] I Sato ldquoOn a structure similar to the almost contact structurerdquoTensor vol 30 no 3 pp 219ndash224 1976

[15] P K Rashevskij ldquoThe scalar field in a stratified spacerdquoTrudy Seminara po Vektornomu i Tenzornomu Analizu s ikhPrilozheniyami k Geometrii Mekhanike i Fizike vol 6 pp 225ndash248 1948

[16] P Libermann ldquoSur les structures presque paracomplexesrdquoComptes Rendus de lrsquoAcademie des Sciences I vol 234 pp 2517ndash2519 1952

[17] E M Patterson ldquoRiemann extensions which have Kahlermetricsrdquo Proceedings of the Royal Society of Edinburgh AMathematics vol 64 pp 113ndash126 1954

[18] S Sasaki ldquoOn differentiable manifolds with certain structureswhich are closely related to almost contact structure Irdquo TheTohoku Mathematical Journal vol 12 pp 459ndash476 1960

[19] S Zamkovoy ldquoCanonical connections on paracontact mani-foldsrdquo Annals of Global Analysis and Geometry vol 36 no 1pp 37ndash60 2009

[20] D E Blair Riemannian Geometry of Contact and SymplecticManifolds vol 23 of Progress in Mathematics BirkhauserBoston Mass USA 2002

[21] V Cruceanu P Fortuny and P M Gadea ldquoA survey onparacomplex geometryrdquoThe Rocky Mountain Journal of Mathe-matics vol 26 no 1 pp 83ndash115 1996

[22] P M Gadea and J M Masque ldquoClassification of almost para-Hermitian manifoldsrdquo Rendiconti di Matematica e delle sueApplicazioni VII vol 7 no 11 pp 377ndash396 1991

[23] M Falcitelli S Ianus and A M Pastore Riemannian Submer-sions and Related Topics World Scientific River Edge NJ USA2004

[24] E G Rio and D N Kupeli Semi-Riemannian Maps andTheir Applications Kluwer Academic Publisher DordrechtTheNetherlands 1999

[25] B Sahin ldquoSemi-invariant submersions from almost HermitianmanifoldsrdquoCanadianMathematical Bulletin vol 54 no 3 2011

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 2: Research Article Paracomplex Paracontact Pseudo-Riemannian …downloads.hindawi.com/archive/2014/616487.pdf · 2019-07-31 · Research Article Paracomplex Paracontact Pseudo-Riemannian

2 Geometry

In Section 3 we define paracomplex paracontact pseudo-Riemannian submersion giving some relevant examples andinvestigate transference of structures on the total manifoldsand base manifolds under such submersions In Section 4curvature relations between total manifolds base manifoldsand fibres are studied

2 Preliminaries

21 Almost Paracontact Manifolds Let 119872 be a (2119899 + 1)-dimensional Riemannianmanifold 120601 a (11)-type tensor field120585 a vector field called characteristic vector field and 120578 a 1-form on 119872 Then (120601 120585 120578) is called an almost paracontactstructure on119872 if

1206012

119883 = 119883 minus 120578 (119883) 120585 120578 (120585) = 1 (1)

and the tensor field 120601 induces an almost paracomplexstructure on the distributionD = ker(120578) [18 19]

119872 is said to be an almost paracontact manifold if it isequipped with an almost paracontact structure Again119872 iscalled an almost paracontact pseudometric manifold if it isendowed with a pseudo-Riemannian metric 119892 of signature(minus minus minus minus⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

(119899-times) + + + +⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

(119899+1)-times)) such that

119892 (120601119883 120601119884) = 119892 (119883 119884) minus 120576120578 (119883) 120578 (119884) forall119883 119884 isin Γ (119879119872)

(2)

where 120576 = 1 or minus1 according to the characteristic vector field120585 is spacelike or timelike It follows that

119892 (120585 120585) = 120576 (3)119892 (120585 119883) = 120576120578 (119883) (4)

119892 (119883 120601119884) = 119892 (120601119883 119884) forall119883 119884 isin Γ (119879119872) (5)

In particular if 119894119899119889119890119909(119892) = 1 then the manifold(1198722119899+1

120601 120585 120578 119892 120576) is called a Lorentzian almost paracontactmanifold

If the metric 119892 is positive definite then the manifold(1198722119899+1

120601 120585 120578 119892) is the usual almost paracontact metricmanifold [14]

The fundamental 2-formΦ on119872 is defined by

Φ (119883 119884) = 119892 (119883 120601119884) (6)

Let 1198722119899+1 be an almost paracontact manifold with thestructure (120601 120585 120578) An almost paracomplex structure 119869 on1198722119899+1

timesR1 is defined by

119869 (119883 119891119889

119889119905) = (120601119883 + 119891120585 120578 (119883)

119889

119889119905) (7)

where119883 is tangent to1198722119899+1 119905 is the coordinate onR1 and 119891is a smooth function on1198722119899+1

An almost paracontact structure (120601 120585 120578) is said to benormal if the Nijenhuis tensor 119873

119869of almost paracomplex

structure 119869 defined as

119873119869(119883 119884) = [119869 119869] (119883 119884) = [119869119883 119869119884] + 119869

2

[119883 119884]

minus 119869 [119869119883 119884] minus 119869 [119883 119869119884] (8)

for any vector fields119883119884 isin Γ(119879119872) vanishes

If 119883 and 119884 are vector fields on1198722119899+1 then we have [18ndash20]

119873119869((119883 0) (119884 0))

= (119873120601(119883 119884) minus 2119889120578 (119883 119884) 120585

(L120601119883120578)119884 minus (L

120601119884120578)119883

119889

119889119905)

(9)

119873119869((119883 0) (0

119889

119889119905)) = minus((L

120585120601)119883 ((L

120585120578)119883)

119889

119889119905)

(10)

where 119873120601is Nijenhuis tensor of 120601L

119883is Lie derivative with

respect to a vector field119883 and119873(1) 119873(2) 119873(3) and 119873(4) aredefined as

119873120601(119883 119884)

= [120601 120601] (119883 119884)

= [120601119883 120601119884] + 1206012

[119883 119884] minus 120601 [120601119883 119884] minus 120601 [119883 120601119884]

(11)

119873(1)

(119883 119884) = 119873120601(119883 119884) minus 2119889120578 (119883 119884) 120585 (12)

119873(2)

(119883 119884) = (L120601119883120578)119884 minus (L

120601119884120578)119883 (13)

119873(3)

(119883) = (L120585120601)119883 (14)

119873(4)

(119883) = (L120585120578)119883 (15)

The almost paracontact structure (120601 120585 120578) is normal if andonly if the four tensors119873(1) 119873(2) 119873(3) and 119873(4) vanish

For an almost paracontact structure (120601 120585 120578) vanishing of119873(1) implies the vanishing of119873(2) 119873(3) and119873(4) Moreover

119873(2) vanishes if and only if 120585 is a killing vector fieldAn almost paracontact pseudometric manifold

(1198722119899+1

120601 120585 120578 119892 120576) is called

(i) normal if119873120601minus 2119889120578 otimes 120585 = 0

(ii) paracontact if Φ = 119889120578(iii) 119870-paracontact if119872 is paracontact and 120585 is killing(iv) paracosymplectic if nablaΦ = 0 which implies nabla120578 = 0

where nabla is the Levi-Civita connection on119872(v) almost paracosymplectic if 119889120578 = 0 and 119889Φ = 0(vi) weakly paracosymplectic if119872 is almost paracosym-

plectic and [119877(119883 119884) 120601] = 119877(119883 119884)120601 minus 120601119877(119883 119884) = 0where 119877 is Riemannian curvature tensor

(vii) para-Sasakian if Φ = 119889120578 and119872 is normal(viii) quasi-para-Sasakian if 119889120601 = 0 and119872 is normal

22 Almost Paracomplex Manifolds A (1 1)-type tensor field119869 on 2119898-dimensional smooth manifold 119872 is said to be analmost paracomplex structure if 1198692 = 119868 and (1198722119898 119869) is calledalmost paracomplex manifold

Geometry 3

An almost paracomplex manifold (119872 119869) is such thatthe two eigenbundles 119879

+

119872 and 119879minus

119872 corresponding torespective eigenvalues +1 and minus1 of 119869 have the same rank[21 22]

An almost para-Hermitianmanifold (119872 119869 119892) is a smoothmanifold endowed with an almost paracomplex structure 119869and a pseudo-Riemannian metric 119892 such that

119892 (119869119883 119869119884) = minus119892 (119883 119884) forall119883 119884 isin Γ (119879119872) (16)

Here the metric 119892 is neutral that is 119892 has signature (119898119898)The fundamental 2-form of the almost para-Hermitian

manifold is defined by

119865 (119883 119884) = 119892 (119883 119869119884) (17)

We have the following properties [21 22]

119892 (119869119883 119884) = minus119892 (119883 119869119884) (18)

119865 (119883 119884) = minus119865 (119884119883) (19)

119865 (119869119883 119869119884) = minus119865 (119883 119884) (20)

3119889119865 (119883 119884 119885)

= 119883 (119865 (119884 119885)) minus 119884 (119865 (119883 119885)) + 119885 (119865 (119883 119884))

minus 119865 ([119883 119884] 119885) + 119865 ([119883 119885] 119884) minus 119865 ([119884 119885] 119883)

(21)

(nabla119883119865) (119884 119885) = 119892 (119884 (nabla

119883119869) 119885) = minus119892 (119885 (nabla

119883119869) 119884) (22)

3119889119865 (119883 119884 119885) = (nabla119883119865) (119884 119885) + (nabla

119884119865) (119885119883)

+ (nabla119885119865) (119883 119884)

(23)

the co-differential (120575119865) (119883) =2119898

sum

119894=1

120576119894(nabla119890119894119865) (119890119894 119883) (24)

An almost para-Hermitian manifold is called

(i) para-Hermitian if 119873119869= 0 equivalently (nabla

119869119883119869)119869119884 +

(nabla119883119869)119884 = 0

(ii) para-Kahler if for any 119883 isin Γ(119879119872) nabla119883119869 = 0 that is

nabla119869 = 0(iii) almost para-Kahler if 119889119865 = 0(iv) nearly para-Kahler if (nabla

119883119869)119883 = 0

(v) almost semi-para-Kahler if 120575119865 = 0(vi) semi-para-Kahler if 120575119865 = 0 and119873

119869= 0

23 Pseudo-Riemannian Submersion Let (119872119898

119892) and(119872119899

119892) be two connected pseudo-Riemannian manifolds ofindices 119904 (0 le 119904 le 119898) and 119904 (0 le 119904 le 119899) respectively with119904 ge 119904

A pseudo-Riemannian submersion is a smooth map 119891

119872119898

rarr 119872119899 which is onto and satisfies the following

conditions [2 3 23 24]

(i) The derivative map 119891lowast119901

119879119901119872 rarr 119879

119891(119901)119872 is

surjective at each point 119901 isin 119872

(ii) The fibres 119891minus1(119902) of 119891 over 119902 isin 119872 are eitherpseudo-Riemannian submanifolds of 119872 ofdimension (119898 minus 119899) and index ] or the degeneratesubmanifolds of 119872 of dimension (119898 minus 119899) andindex ] with degenerate metric 119892

|119891minus1(119902)

of type(0 0 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

120583-times minus minus minus minus⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

]-times + + + +⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

(119898minus119899minus120583minus])-times)) where

120583 = dim(V119901capH119901) and ] = 119904 minus 119904 = index of 119892

|119891minus1(119902)

(iii) 119891lowastpreserves the length of horizontal vectors

We denote the vertical and horizontal projections of avector field 119864 on 119872 by 119864V (or by V119864) and 119864

ℎ (or by ℎ119864)respectively A horizontal vector field 119883 on 119872 is said to bebasic if 119883 is 119891-related to a vector field 119883 on119872 Thus everyvector field119883 on119872 has a unique horizontal lift119883 on119872

Lemma 1 (see [1 23]) If 119891 119872 rarr 119872 is a pseudo-Riemannian submersion and119883 119884 are basic vector fields on119872that are 119891-related to the vector fields 119883 119884 on119872 respectivelythen one has the following properties

(i) 119892(119883 119884) = 119892(119883 119884) ∘ 119891(ii) ℎ[119883 119884] is a vector field and ℎ[119883 119884] = [119883 119884] ∘ 119891(iii) ℎ(nabla

119883119884) is a basic vector field 119891-related to nabla

119883119884 where

nabla and nabla are the Levi-Civita connections on119872 and119872respectively

(iv) [119864 119880] isin V for any vector field 119880 isin V and for anyvector field 119864 isin Γ(119879119872)

A pseudo-Riemannian submersion 119891 119872 rarr 119872

determines tensor fieldsT andA of type (1 2) on119872 definedby formulas [1 2 23]

T (119864 119865) = T119864119865 = ℎ (nablaV119864V119865) + V (nablaV119864ℎ119865) (25)

A (119864 119865) = A119864119865 = V (nabla

ℎ119864ℎ119865) + ℎ (nabla

ℎ119864V119865)

for any 119864 119865 isin Γ (119879119872)

(26)

Let 119883 119884 be horizontal vector fields and let 119880 119881 bevertical vector fields on119872 Then one has

T119880119883 = V (nabla

119880119883) T

119880119881 = ℎ (nabla

119880119881) (27)

nabla119880119883 = T

119880119883 + ℎ (nabla

119880119883) (28)

T119883119865 = 0 T

119864119865 = TV119864119865 (29)

nabla119880119881 = T

119880119881 + V (nabla

119880119881) (30)

A119883119884 = V (nabla

119883119884) A

119883119880 = ℎ (nabla

119883119880) (31)

nabla119883119880 = A

119883119880 + V (nabla

119883119880) (32)

A119880119865 = 0 A

119864119865 = A

ℎ119864119865 (33)

nabla119883119884 = A

119883119884 + ℎ (nabla

119883119884) (34)

4 Geometry

ℎ (nabla119880119883) = ℎ (nabla

119883119880) = A

119883119880 (35)

A119883119884 =

1

2V [119883 119884] (36)

A119883119884 = minusA

119884119883 (37)

T119880119881 = T

119881119880 (38)

for all 119864 119865 isin Γ(119879119872)MoreoverT

119880119881 coincideswith second fundamental form

of the submersion of the fibre submanifoldsThe distributionH is completely integrable In view of (37) and (38) A isalternating on the horizontal distribution andT is symmetricon the vertical distribution

3 Paracomplex Paracontact Pseudo-Riemannian Submersions

In this section we introduce the notion of pseudo-Riemannian submersion from almost paracomplex mani-folds onto almost paracontact pseudometric manifolds illus-trate examples and study the transference of structures ontotal manifolds and base manifolds

Definition 2 Let (1198722119898 119869 119892) be an almost para-Hermitianmanifold and let (1198722119899+1 120601 120585 120578 119892) be an almost paracontactpseudometric manifold

A pseudo-Riemannian submersion 119891 119872 rarr 119872 is calledparacomplex paracontact pseudo-Riemannian submersion ifthere exists a 1-form 120578 on119872 such that

119891lowast∘ 119869 = 120601 ∘ 119891

lowast+ 120578 otimes 120585 (39)

Since for each 119901 isin 119872119891lowast119901

is a linear isometry betweenhorizontal spacesH

119901and tangent spaces119879

119891(119901)119872 there exists

an induced almost paracontact structure (120601ℎ

120578ℎ

120585ℎ

119892) on(2119899 + 1)-dimensional horizontal distribution H such that120601ℎ

|

Dℎbehave just like the fundamental collineation of almost

paracomplex structure 119869 on ker 120578ℎ = Dℎ

and 120601ℎ

Dℎ

rarr Dℎ

is an endomorphism such that 120601ℎ

= 119869|ker 120578ℎ

and the rank of

120601ℎ

= 2119899 where dim(Dℎ

) = 2119899It follows that for any 119883ℎ isin D

ℎ

120578ℎ(119883ℎ) = 0 whichimplies that 1198692

|

Dℎ(119883ℎ

) = (120601ℎ

)2

(119883ℎ

) = 119883ℎ for any 119883ℎ isin D

ℎ

andH = Dℎ

oplus 120585ℎ

[18]

Definition 3 (see [25]) A pseudo-Riemannian submersion119891 119872 rarr 119872 is called semi-119869-invariant submersion if thereis a distributionD

1sube ker119891

lowastsuch that

ker119891lowast= D1oplusD2 (40)

119869 (D1) = D

1 119869 (D

2) sube (ker119891

lowast)perp

(41)

whereD2is orthogonal complementary toD

1in ker119891

lowast

Proposition 4 Let119891 1198722119898 rarr 1198722119899+1 be a paracomplex par-

acontact pseudo-Riemannian submersion and let the fibres of119891 be pseudo-Riemannian submanifolds of119872 Then the fibres119891minus1

(119902) 119902 isin 119872 are semi-119869-invariant submanifolds of 119872 ofdimension (2119898 minus 2119899 minus 1)

Proof Let 119880 isinV Then

119891lowast(119869119880) = 120601 (119891

lowast(119880)) + 120578 (119880) 120585

997904rArr 119891lowast119869 (119880) minus 120578 (119880) 120585

ℎ

= 0

(42)

where 119891lowast120585ℎ

= 120585Thus we have

119869 (119880) minus 120578 (119880) 120585ℎ

= 120601 (119880) for some 120601 (119880) isinV (43)

By (19) we get 119892(120585ℎ

119869(120585ℎ

)) = 0 = 119892(120585 119891lowast(119869(120585ℎ

))) = 0As 119892 is nondegenerate on119872 we have

119891lowast(119869 (120585ℎ

)) = 0 that is 119869 (120585ℎ

) isinV (44)

Taking 119880 = 119869120585ℎ

in (43) we obtain

120585ℎ

minus 120578 (119869120585ℎ

) 120585ℎ

= 120601(119869120585ℎ

) (45)

Since fibre 119891minus1(119902) is an odd dimensional submanifold thereexists an associated 1-form 120578

V which is restriction of 120578 on fibresubmanifold 119891minus1(119902) 119902 isin 119872 and a characteristic vector field120585V= 119869120585ℎ

such that 120601(120585V) = 0 So we have 120578V(120585

V) = 1

Let us put ker 120578V = D1andD

2= 120585

V

Then ker119891lowast= D1oplus D2and 119869(D

1) = D

1 119869(D

2) =

119869120585V = 120585

ℎ

sube (ker119891lowast)perp

Hence the fibres 119891minus1(119902) are semi-119869-invariant submani-folds of119872

Corollary 5 Let 119891 1198722119898

rarr 1198722119899+1 be a paracomplex

paracontact pseudo-Riemannian submersion and let the fibresof 119891 be pseudo-Riemannian submanifolds of 119872 Then thefibres 119891

minus1

(119902) are almost paracontact pseudometric mani-folds with almost paracontact pseudo-Riemannian structures(120601

V 120585

V 120578

V 119892

V) 119902 isin 119872 where 120585

V= 119869(120585

ℎ

) 120578V = 120578|V and

119892V= 119892

Proof Since 119891minus1(119902) are semi-119869-invariant submanifolds of119872of odd dimension 2119903 + 1 = 2119898 minus 2119899 minus 1 (39) implies

119869 (119880) = 120601V119880 + 120578

V(119880) 120585ℎ

(46)

for any 119880 isinV

Geometry 5

On operating 119869 on both sides of the above equation weget

119880 = 120601V(120601

V(119880)) + 120578

V(120601

V(119880)) 120585

ℎ

+ 120578V(119880) 120585

V (47)

where 119869(120585ℎ

) = 120585V

Equating horizontal and vertical components we have

119880 = 120601V(120601

V(119880)) + 120578

V(119880) 120585

V 120578

V∘ 120601

V(119880) = 0

997904rArr (120601V)2

(119880) = 119880 minus 120578V(119880) 120585

V 120578

V∘ 120601

V= 0

120601V(120585

V) = 0 120578

V(120585

V) = 1

(48)

Hence (120601V 120585

V 120578

V 119892

V) is almost paracontact pseudometric

structure on the fibre 119891minus1(119902) 119902 isin 119872

Proposition 6 Let 119891 1198722119898

rarr 1198722119899+1 be a paracomplex

paracontact pseudo-Riemannian submersion and let the fibresof 119891 be pseudo-Riemannian submanifolds of 119872 Let 119883 119884 bebasic vector fields 119891-related to 119883 119884 respectively Let 120578 and 120578be 1-forms on the total manifold119872 and the base manifold119872respectively Then one has the following

(i) The characteristic vector field 119869120585ℎ

is a vertical vectorfield

(ii) 119891lowastlowast120578 = 120578ℎ where 119891lowast

lowast120578 is pullback of 120578 through 119891

lowast

(iii) 120578ℎ(119880) = 0 for any vertical vector field 119880(iv) 120578V(119883) = 0 for any horizontal vector field119883

Remark 7 Results (ii) and (iv) are analogue version of results(i) and (iii) of Proposition 4 of [13]

Proof (i) By Corollary 5 (120601V 120585

V 120578

V 119892

V) is almost paracontact

pseudometric structure on 119891minus1(119902) We have

0 = 119892 (120585ℎ

119869120585ℎ

) = 119892(119891lowast(120585ℎ

) 119891lowast(119869120585ℎ

))

= 119892(120585 119891lowast(119869120585ℎ

))

(49)

Now

119891lowast(119869120585ℎ

) = 120601 ∘ 119891lowast120585ℎ

+ 120578 (120585ℎ

) 120585 = 120578 (120585ℎ

) 120585 (50)

so we have

0 = 119892(120585 120578 (120585ℎ

) 120585) = 120578 (120585ℎ

)119892 (120585 120585) = 120578 (120585ℎ

) (51)

Thus 119891lowast(119869120585ℎ

) = 0Hence 119869120585

ℎ

is a vertical vector field(ii) Since 119891 119872

2119898

rarr 1198722119899+1 is smooth submersion

120578ℎ

= 120578|H

is restriction of 120578 on the horizontal distribution

H and 119891lowast119901

H119901rarr 119879119891(119901)

119872 is a linear isometry for any119883119901isinH119901 we get

120578ℎ

119901(119883119901) = 120576119892

119901(120585ℎ

119901 119883119901) = 119892

119891(119901)(119891lowast119901120585ℎ

119901 119891lowast119901119883119901)

= 119892119891(119901)

(120585119891(119901)

119883119891(119901)

) = 120578119891(119901)

(119883119891(119901)

) = 119891lowast

lowast120578119901(119883119901)

(52)

Hence pullback 119891lowastlowast120578 = 120578ℎ

Results (iii) and (iv) immediately follow from the previ-ous results

Example 8 Let (R42 119869 119892) be a paracomplex pseudometric

manifold and let (R31 120601 120585 120578 119892) be an almost paracontact

pseudometric manifoldDefine a submersion 119891 R4

2 (1199091 1199092 1199101 1199102)119905

rarr

R31 (119906 V 119908)119905 by

119891 ((1199091 1199092 1199101 1199102)119905

) 997891997888rarr (1199091+ 1199092+ 31199101+ 21199102

31199091+ 21199092+ 1199101+ 1199102

51199091+ 31199092+ 51199101+ 31199102)119905

(53)

Then the kernel of 119891lowastis

V = ker119891lowast= Span119881

1=

120597

1205971199091

minus 2120597

1205971199092

minus120597

1205971199101

+ 2120597

1205971199102

(54)

which is the vertical distribution admitting one lightlikevector field that is fibre is degenerate submanifold of R4

2

The horizontal distribution is

H = (ker119891lowast)perp

= Span1198831=

120597

1205971199091

minus120597

1205971199101

1198832=

120597

1205971199092

+ 2120597

1205971199101

1198833= 2

120597

1205971199101

+120597

1205971199102

(55)

For any real 119896 the horizontal characteristic vector field 120585ℎ

isgiven by

120585ℎ

= 119896120597

1205971199091

minus (2119896 minus1

3)

120597

1205971199092

minus (119896 minus 1)120597

1205971199101

+ (2119896 minus5

3)

120597

1205971199102

(56)

which is 119891-related to the characteristic vector field 120585 = 120597120597119908Moreover there exists one form 120578 = 5119889119909

1+3119889119909

2+51198891199101+

31198891199102on (R4

2 119869 119892) such that the submersion satisfies (39)

Example 9 Let (R63 119869 119892) be an almost paracomplex pseudo-

Riemannian manifold and let (R31 120601 120585 120578 119892) be an almost

6 Geometry

paracontact pseudo-Riemannian manifold Consider a sub-mersion 119891 R6

3 (1199091 1199092 1199093 1199101 1199102 1199103)119905

rarr R31 (119906 V 119908)119905

defined by

119891 ((1199091 1199092 1199093 1199101 1199102 1199103)119905

)

997891997888rarr (1199091+ 1199092

radic2

1199101+ 1199102

radic2

1199102+ 1199103

radic2

)

119905

(57)

Then there exists one form 120578 = (1198891199092+ 1198891199093)radic2 on (R6

3 119869 119892)

such that (39) is satisfied The kernel of 119891lowastis

V = ker119891lowast

= Span1198811=

120597

1205971199091

minus120597

1205971199092

1198812=

120597

1205971199101

minus120597

1205971199102

+120597

1205971199103

1198813=

120597

1205971199093

(58)

which is vertical distribution admitting non-lightlike vectorfields that is the fibre is nondegenerate submanifold of(R63 119869 119892)The horizontal distribution is

H = Span1198831=

120597

1205971199091

+120597

1205971199092

1198832= minus

120597

1205971199101

+120597

1205971199103

1198833=

120597

1205971199101

+120597

1205971199102

(59)

Example 10 Let (R42 119869 119892) be a paracomplex pseudometric

manifold and let (R31 120601 120585 120578 119892) be an almost paracontact

pseudometric manifoldConsider a submersion 119891 R4

2 (1199091 1199092 1199101 1199102)119905

rarr R31

(119906 V 119908)119905 defined by

119891 ((1199091 1199092 1199101 1199102)119905

) 997891997888rarr (1199091 1199101 1199102)119905

(60)

Then the kernel of 119891lowastis

V = ker119891lowast= Span119881

1=

120597

1205971199092

(61)

which is the vertical distribution and the restriction of 119892 tothe fibres of 119891 is nondegenerate

The horizontal distribution is

H = (ker119891lowast)perp

= Span119883 =120597

1205971199091

119884 =120597

1205971199101

120585ℎ

=120597

1205971199102

(62)

The characteristic vector field 120585 = 120597120597119908 on R31has unique

horizontal lift 120585ℎ

which is the characteristic vector field onhorizontal distributionH of R4

2

We also have

119892 (119883119883) = 119892 (119891lowast119883119891lowast119883) = minus1

119892 (119884 119884) = 119892 (119891lowast119884 119891lowast119884) = 1

119892 (120585ℎ

120585ℎ

) = 119892(119891lowast120585ℎ

119891lowast120585ℎ

) = 119892 (120585 120585) = 1

(63)

Thus the smooth map 119891 is a pseudo-Riemannian submer-sion

Moreover we obtain that there exists a 1-form 120578 = 1198891199092on

R42such that 120578(119869120585

ℎ

) = 1 120578(120585ℎ

) = 0 and the map 119891 satisfies

119891lowast119869119883 = 120601119891

lowast119883 + 120578 (119883) 120585

119891lowast119869119884 = 120601119891

lowast119884 + 120578 (119884) 120585

119891lowast119869120585ℎ

= 120601119891lowast120585ℎ

+ 120578 (120585ℎ

) 120585

(64)

Hence the map 119891 is a paracomplex paracontact pseudo-Ri-emannian submersion from R4

2on to R3

1

Proposition 11 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion and let the fibresof 119891 be pseudo-Riemannian submanifolds of 119872 Let 119883 119884 bebasic vector fields 119891-related to119883 119884 respectively Then 119869(119883) minus120576119892(119883 120585

ℎ

)120585ℎ

is 119891-related to 120601119883

Proof Since119883 is 119891-related to vector field119883 on119872 we have

120578 (119883) = 120578V+ 120578ℎ

(119883) = 0 + 120578ℎ

(119883) = 120576119892 (119883 120585ℎ

)

997904rArr 119891lowast(119869119883) = 120601119883 + 120578

ℎ

(119883) 120585

997904rArr 119891lowast119869119883 minus 120576119892 (119883 120585

ℎ

) 120585ℎ

= 120601119883

(65)

Hence 119869(119883) minus 120576119892(119883 120585ℎ

)120585ℎ

is 119891-related to 120601119883

Proposition 12 Let 119891 119872 rarr 119872 be a paracomplex para-contact pseudo-Riemannian submersion and let the fibres of 119891be pseudo-Riemannian submanifolds of 119872 Let V and H bethe vertical and horizontal distributions respectively If 120585

ℎ

isthe basic characteristic vector field of horizontal distribution119891-related to the characteristic vector field 120585 of base manifoldthen

(i) 119869V sub DVoplus 119869120585ℎ

oplus 120585ℎ

(ii) 119869H sub Dℎ

oplus 120585ℎ

oplus 119869120585ℎ

Proof (i) Let 119880 isin V Then 119880 = 119886119880|DV + 119887119869120585

ℎ

for 119886 119887 isin

119862infin

(119872) as 119869120585ℎ

= 120585Vis characteristic vector field on odd

Geometry 7

dimensional fibre submanifold 119891minus1(119902) of119872 119902 isin 119872 We get

119869119880 = 119886119869119880|DV + 119887119869

2

120585ℎ

= 119886119869119880|DV + 119887120585

ℎ

isinV oplus 120585ℎ

997904rArr 119869V subV oplus 120585ℎ

(66)

Again let 119881 isinV oplus 120585ℎ

Then 119881 = 119886119881|DV + 119887119869120585

ℎ

+ 119888120585ℎ

where

120578V(119881|DV ) = 0 D

V= ker 120578V 119886119881

|DV + 119887119869120585

ℎ

isin V and 119886 119887 119888 isin119862infin

(119872) We have

119869119881 = 119886119869119881|DV + 119887120585

ℎ

+ 119888119869120585ℎ

= (119886119869119881|DV + 119888119869120585

ℎ

)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

isinV

+ 119887120585ℎ

⏟⏟⏟⏟⏟⏟⏟

isin120585

ℎ

isinV oplus 120585ℎ

(67)

Now by (39) we get

119891lowast119869119881 = 120601 (119891

lowast119881) + 120578 (119891

lowast(119881)) 120585

= 119888 120601 (119891lowast120585ℎ

) + 120578 (120585) 120585

= 119888120585 isin 120585 sube 119869V

(68)

We get 119869119881 notinVHence 119869V subV oplus 120585

ℎ

that is 119869V sub DVoplus 119869120585ℎ

oplus 120585ℎ

(ii) Let 119883 = 119886119883

|

Dℎ+ 119887120585ℎ

isin H where H = Dℎ

oplus

120585ℎ

ker 120578ℎ = Dℎ

and 119886 119887 isin 119862infin(119872) Then

119869119883 = 119886119869119883|

Dℎ+ 119887119869120585ℎ

isinH oplus 119869120585ℎ

(69)

which implies that 119869H subH oplus 119869120585ℎ

Again let119884 isinHoplus119869120585

ℎ

Then119884 = 119886119884|

Dℎ+119887120585ℎ

+119888119869120585ℎ

notinHfor 119886 119887 119888 isin 119862infin(119872) We have

119869 119884 = 119886119869119884|

Dℎ+ 119887119869120585ℎ

+ 1198881198692

120585ℎ

= 119886119869119884|

Dℎ+ 119887119869120585ℎ

+ 119888120585ℎ

= 119885 + 119887119869120585ℎ

isinH oplus 119869120585ℎ

for some 119885 = 119886119869119884|

Dℎ+ 119888120585ℎ

isinH

(70)

We obtain 119869 119884 notinHHence 119869H sub H oplus 119869(120585

ℎ

) that is 119869H sub Dℎ

oplus 120585ℎ

oplus

119869120585ℎ

Example 13 Let (R63 119869 119892) be an almost paracomplex pseudo-

Riemannian manifold and let (R31 120601 120585 120578 119892) be an almost

paracontact pseudo-Riemannian manifold Consider a sub-mersion 119891 R6

3 (1199091 1199092 1199093 1199101 1199102 1199103)119905

rarr R31 (119906 V 119908)119905

defined by

119891 ((1199091 1199092 1199093 1199101 1199102 1199103)119905

) 997891997888rarr (1199091+ 1199092

radic2

1199101+ 1199102

radic2

1199103)

119905

(71)

Then the kernel of 119891lowastis

V = ker119891lowast

= Span1198811=

120597

1205971199091

minus120597

1205971199092

1198812=

120597

1205971199101

minus120597

1205971199102

120585V=

120597

1205971199093

(72)

which is the vertical distribution and the restriction of 119892 tothe fibres of 119891 is nondegenerate

The horizontal distribution is

H = (ker119891lowast)perp

= Span1198831=

120597

1205971199091

+120597

1205971199092

1198832=

120597

1205971199101

+120597

1205971199102

120585ℎ

=120597

1205971199103

(73)

The characteristic vector field 120585 = 120597120597119908 on R31has unique

horizontal lift 120585ℎ

which is the characteristic vector field onthe horizontal distributionH of R6

3

We also have

119892 (1198831 1198831) = 119892 (119891

lowast1198831 119891lowast1198831) = minus2

119892 (1198832 1198832) = 119892 (119891

lowast1198832 119891lowast1198832) = 2

119892 (120585ℎ

120585ℎ

) = 119892(119891lowast120585ℎ

119891lowast120585ℎ

) = 119892 (120585 120585) = 1

(74)

Thus the smooth map 119891 is a pseudo-Riemannian submer-sion

Also we obtain that there exists a 1-form 120578 = 1198891199093on R63

such that 120578(119869120585ℎ

) = 1 120578(120585ℎ

) = 0 and the map 119891 satisfies

119891lowast1198691198831= 120601119891lowast1198831+ 120578 (119883

1) 120585

119891lowast1198691198832= 120601119891lowast1198832+ 120578 (119883

2) 120585

119891lowast119869120585ℎ

= 120601119891lowast120585ℎ

+ 120578 (120585ℎ

) 120585

(75)

Hence the map 119891 is a paracomplex paracontact pseudo-Riemannian submersion from R6

3onto R3

1

Moreover we observe that for this submersion119891 we have

119869V subV oplus 120585ℎ

119869H subH oplus 119869120585ℎ

(76)

which verifies Proposition 12

8 Geometry

Proposition 14 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion and let the fibresof 119891 be pseudo-Riemannian submanifolds of 119872 Let 119883 119884 bebasic vector fields 119891-related to 119883 119884 respectively Let 119865 and Φbe the second fundamental forms and let nabla and nabla be the Levi-Civita connection on the total manifold119872 and base manifold119872 respectively Then one has

(i) 119891lowast((nabla119883119869)119884) = (nabla

119883120601)119884 + 120576119892(119884 nabla

119883120585)120585 + 120578(119884)nabla

119883120585

(ii) 119865 = 119891lowastlowastΦ + 120576120578 otimes 120578

(iii) 119891lowast((nabla119883119865)(119884 119885)) = (nabla

119883Φ)(119884 119885) + 120578(119884)119892(119885 nabla

119883120585) +

120578(119885)119892(119884 nabla119883120585)

Proof (i) In view of Definition 2 and Proposition 11 we have

119891lowast((nabla119883119869) 119884) = 119891

lowast(nabla119883(119869 119884) minus 119869 (nabla

119883119884))

= nabla119883(119891lowast(119869 119884)) minus 119891

lowast(119869 (nabla119883119884))

= nabla119883(120601119884) + nabla

119883(120578 (119884) 120585) minus 120601 (nabla

119883119884)

minus 120578 (nabla119883119884) 120585

= (nabla119883120601)119884 + nabla

119883(120576119892 (119884 120585) 120585) minus 120578 (nabla

119883119884) 120585

= (nabla119883120601)119884 + 120576119892 (nabla

119883119884 120585) 120585 + 120576119892 (119884 nabla

119883120585) 120585

+ 120576119892 (119884 120585) nabla119883120585 minus 120576119892 (nabla

119883119884 120585) 120585

= (nabla119883120601)119884 + 120576119892 (119884 nabla

119883120585) 120585 + 120578 (119884) nabla

119883120585

(77)

(ii) Since119891lowastlowastΦ is pullback ofΦ through the linearmap119891

lowast

we get

119891lowast

lowastΦ(119883 119884) = Φ (119883 119884) ∘ 119891 = 119892 (119883 120601119884) ∘ 119891

= 119892 (119883 119869 119884) minus 120576120578 (119884) 120578 (119883)

= 119865 (119883 119884) minus 120576120578 (119883) 120578 (119884)

(78)

which implies 119865 = 119891lowastlowastΦ + 120576120578 otimes 120578

(iii) By (23) we have

119891lowast((nabla119883119865) (119884 119885)) = 119892 (119891

lowast(119884) 119891

lowast((nabla119883119869)119885)) (79)

Now using (i) in the above equation we get (iii)

Theorem 15 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion and let the fibresof 119891 be pseudo-Riemannian submanifolds of 119872 Let 119883 119884 bebasic vector fields 119891-related to 119883 119884 respectively If the totalspace is para-Hermitian manifold then the almost paracontactstructure of base space is normal

Moreover if the almost paracontact structure of base spaceis normal then the Nijenhuis tensor of total space is vertical

Proof The Nijenhuis tensors 119873119869and 119873

120601of almost para-

complex structure 119869 and almost paracontact structure 120601 arerespectively defined by (8) and (11)

Using Definition 2 and properties of Sections 21 and 22we get the following identity

119891lowast(119873119869(119883 119884)) = 119873

(1)

(119883 119884) + 2119889120578 (120601119883 119884) 120585

minus 2119889120578 (120601119884119883) 120585

+ 2120578 (119883) 119889120578 (120585 119884) 120585 minus 2120578 (119884) 119889120578 (120585 119883) 120585

minus 120578 (119884)119873(3)

(119883) + 120578 (119883)119873(3)

(119884)

(80)

Using (12) (13) (14) and (15) (80) reduces to

119891lowast(119873119869(119883 119884)) = 119873

(1)

(119883 119884) + 119873(2)

(119883 119884) 120585

+ 120578 (119883)119873(4)

(119884) 120585

minus 120578 (119884)119873(4)

(119883) 120585 minus 120578 (119884)119873(3)

(119883)

+ 120578 (119883)119873(3)

(119884)

(81)

Since 119873119869(119883 119884) = 0 it follows from (81) that

tensors 119873(1) 119873(2) 119873(3) and 119873(4) vanish togetherHence the almost paracontact structure of base space is

normalConversely let the almost paracontact structure of the

base space be normalThen (81) implies that 119891

lowast(119873119869(119883 119884)) = 0

Hence119873119869(119883 119884) is vertical

Corollary 16 Let119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion and let the fibres of 119891 bepseudo-Riemannian submanifolds of 119872 Let 119883 119884 be basicvector fields 119891-related to 119883 119884 respectively Let the total spacebe para-Hermitian manifold and119873(1) vanishes Then the basespace is paracontact pseudometric manifold if and only if 120585 iskilling

Proof Let the total space be para-Hermitian and 119873(1) van-ishes Then from (80) we have

0 = 2119889120578 (120601119883 119884) 120585 minus 2119889120578 (120601119884119883) 120585 + 2120578 (119883) 119889120578 (120585 119884) 120585

minus 2120578 (119884) 119889120578 (120585 119883) 120585 minus 120578 (119884) (L120585120601)119883 + 120578 (119883) (L

120585120601)119884

(82)

If 120585 is killing then we have L120585120601 = 0 It immediately follows

from (82) that

119889120578 (120601119883 119884) minus 120578 (120601119884119883) + 120578 (119883) 119889120578 (120585 119884)

minus 120578 (119884) 119889120578 (120585 119883) = 0

(83)

In view of (6) and (7) the above equation gives 119889120578 = ΦConversely let the base space be paracontact Then 119889120578 =

ΦUsing (6) (7) and (82) we getL

120585120601 = 0

Hence the characteristic vector field 120585 is killing

Geometry 9

Theorem 17 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion and let the fibres of 119891 bepseudo-Riemannian submanifolds of 119872 Let 119883 119884 be basicvector fields 119891-related to 119883 119884 respectively If the total spaceis para-Kahler then the base space is paracosymplectic Theconverse is true if nabla

119883119869 is vertical

Proof We have for any 119883119884 isin Γ(119879119872) (nabla119883120601)119884 = 0 which

gives 119892(119885 (nabla119883120601)119884) = 0 for any 119885 isin Γ(119879119872)

From Proposition 14 we have

119892 (119885 (nabla119883120601)119884) + 120576120578 (119884) (nabla

119883120578)119885 + 120576120578 (119885) (nabla

119883120578) 119884 ∘ 119891

= 119892 (119885 (nabla119883119869) 119884)

(84)

Let nabla 119869 = 0 that is the total space is para-KahlerThen from(84) we obtain nabla120601 = 0 and nabla120578 = 0 Hence the base space isparacosymplectic

Again let (nabla119883120601)119884 = 0 and nabla120578 = 0 Then 119892(119885 (nabla

119883119869)119884) =

0 which implies that (nabla119883119869)119884 is a vertical vector field

Theorem 18 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion and let the fibres of 119891 bepseudo-Riemannian submanifolds of 119872 Let 119883 119884 and 119885 bebasic vector fields 119891-related to 119883 119884 and 119885 respectively If120578(119883)119885(120578(119884))+120578(119884)119883(120578(119885))minus120578(119883)119884(120578(119885)) = 0 then the totalspace is almost para-Kahler if and only if the base space119872 isan almost paracosymplectic manifold

Proof We have the following equation

3119889119865 (119883 119884 119885)

= 3 (119891lowast

lowast119889Φ) (119883 119884 119885) + 2120576120578 (119885) 119889120578 (119883 119884)

minus 2120576120578 (119884) 119889120578 (119883 119885)

+ 2120576120578 (119883) 119889120578 (119884 119885) + 2120576120578 (119883)119885 (120578 (119884))

+ 2120576120578 (119884)119883 (120578 (119885))

minus 2120576120578 (119883)119884 (120578 (119885))

(85)

If 119889120578 = 0 119889Φ = 0 and 120578(119883)119885(120578(119884)) + 120578(119884)119883(120578(119885)) minus

120578(119883)119884(120578(119885)) = 0 then from (85) we have 119889119865 = 0 Hencethe total space is almost para-Kahler

Conversely let 119889119865 = 0 and 120578(119883)119885(120578(119884))+120578(119884)119883(120578(119885))minus120578(119883)119884(120578(119885)) = 0

By using the above equation in (85) we have 119889120578 = 0 and119889Φ = 0

Hence the base space is almost paracosymplectic

Nowwe investigate the properties of fundamental tensorsT andA of a pseudo-Riemannian submersion

Lemma 19 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometric manifold119872 and

let the fibres of 119891 be pseudo-Riemannian submanifolds of 119872Then for any horizontal vector fields119883 119884 and for any verticalvector fields 119880 119881 on119872 one has

(i) A119883(119869 119884) = 119869(A

119883119884)

(ii) A119869119883(119884) = 119869(A

119883119884)

(iii) T119880(119869 119881) = 119869(T

119880119881)

(iv) T119869119880119881 = 119869(T

119880119881)

Proof The proof follows using similar steps as in Lemmas 3and 4 of [13] so we omit it

Lemma 20 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometric manifold119872 andlet the fibres of 119891 be pseudo-Riemannian submanifolds of 119872Then for any vector fields 119864 119865 on119872 one has

(i) A119864(119869119865) = 119869(A

119864119865)

(ii) T119864(119869119865) = 119869(T

119864119865)

Proof The proof follows from (37) and (38)

Theorem 21 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometric manifold119872 andlet the fibres of 119891 be pseudo-Riemannian submanifolds of 119872Then the horizontal distribution is integrable

Proof For any vertical vector field 119880 we have

119892 (119869 (A119883119884) 119880) = 119892 (A

119883119869 119884119880)

= minus119892 (119869119884A119883119880)

= minus119892 (119869119884 ℎ (nabla119880119883))

= 119892 (119884 ℎ (119869 (nabla119880119883)))

= 119892 (119884 ℎ (minusnabla119880119869)119883

+nabla119880(119869119883))

= 119892 (119884 ℎ nabla119880(119869119883)) = 119892 (119884A

119869119883119880)

= minus119892 (A119869119883119884119880) = minus119892 (119869 (A

119883119884) 119880)

(86)

Thus 119892(119869(A119883119884) 119880) = 0 which is true for all119883 and 119884

SoA119883119884 = 0

Hence the horizontal distribution is integrable

Theorem 22 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometric manifold119872 andlet the fibres of 119891 be pseudo-Riemannian submanifolds of 119872Then the submersion is an affine map onH

10 Geometry

Proof The second fundamental form of 119891 is defined by

(nabla119891lowast) (119864 119865) = (nabla

119891

119864119891lowast(119865)) ∘ 119891 minus 119891

lowast(nabla119864119865) (87)

where 119864 119865 isin Γ(119879119872) and nabla119891 is pullback connection of Levi-Civita connection nabla on119872 with respect to 119891

We have for any119883119884 isinH

(nabla119891lowast) (119883 119884) = (nabla

119891

119883

119891lowast(119884)) ∘ 119891 minus 119891

lowast(nabla119883119884) (88)

By using Lemma 1 we have 119891lowast(ℎ(nabla119883119884)) = (nabla

119883119884) ∘ 119891 which

implies nabla119891lowast= 0

Hence the submersion 119891 is an affine map onH

Theorem 23 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Hermitian man-ifold119872 onto an almost paracontact pseudometric manifold119872and let the fibres of 119891 be pseudo-Riemannian submanifolds of119872 Then the submersion is an affine map on V if and only ifthe fibres of 119891 are totally geodesic

Proof We have for any 119880119881 isinV

(nabla119891lowast) (119880 119881) = minus119891

lowast(ℎ (nabla119880119881)) (89)

which in view of (27) gives

(nabla119891lowast) (119880 119881) = minus119891

lowast(T119880119881) (90)

Let the fibres of 119891 be totally geodesic Then T = 0Consequently from the above equation we have nabla119891

lowast= 0

Thus the map 119891 is affine onVConversely let the submersion 119891 be an affine map onV

Then nabla119891lowast= 0 which impliesT = 0

Hence the fibres of 119891 are totally geodesic

Theorem 24 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Hermitian man-ifold119872 onto an almost paracontact pseudometric manifold119872and let the fibres of 119891 be pseudo-Riemannian submanifoldsof 119872 Then the submersion is an affine map if and only ifℎ(nabla119864ℎ119865)+A

ℎ119864V119865+TV119864V119865 is 119891-related to nabla

119883119884 for any 119864 119865 isin

Γ(119879119872)

Proof For any 119864 119865 isin Γ(119879119872) with 119891lowastℎ119864 = 119883 ∘ 119891 and 119891

lowastV119865 =

119884 ∘ 119891 we have

(nabla119891lowast) (119864 119865) = (nabla

119891lowastℎ119864(119891lowastℎ119865)) ∘ 119891 minus 119891

lowast(ℎ (nabla119864119865))

= (nabla119883119884) ∘ 119891 minus 119891

lowast(ℎ (nablaℎ119864ℎ119865 + nabla

ℎ119864V119865

+nablaV119864ℎ119865 + nablaV119864ℎ119865))

(91)

By using (27) and (31) in the above equation we have

(nabla119891lowast) (119864 119865) = (nabla

119883119884) ∘ 119891 minus 119891

lowast(ℎ (nabla119864ℎ119865) +A

ℎ119864V119865

+TV119864V119865) (92)

Let the submersion map be affine Then for any 119864 119865 isin

Γ(119879119872) (nabla119891lowast)(119864 119865) = 0 Equation (92) implies (nabla

119883119884) ∘ 119891 =

119891lowast(ℎ(nabla119864ℎ119865) +A

ℎ119864V119865 +TV119864ℎ119865)

Conversely let ℎ(nabla119864119865) + A

ℎ119864V119865 + TV119864ℎ119865 be 119891-related

to nabla119883119884 for any 119864 119865 isin Γ(119879119872) Then from (92) we have

(nabla119891lowast)(119864 119865) = 0Hence the submersion map 119891 is affine

4 Curvature Properties

In this section the paraholomorphic bisectional curvaturesand paraholomorphic sectional curvatures of total mani-fold base manifold and fibres of paracomplex paracontactpseudo-Riemannian submersion and their curvature proper-ties are studied

Let 119891 119872 rarr 119872 be a paracomplex paracontact pseudo-Riemannian submersion from an almost para-Hermitianmanifold (119872 119869 119892) onto an almost paracontact pseudometricmanifold (119872 120601 120585 120578 119892)

Suppose that the vector fields 119864 119865 span the 2-dimensional plane at point 119901 of 119872 and let R be theRiemannian curvature tensor of 119872 The paraholomorphicbisectional curvature 119861(119864 119865) of 119872 for any pair of nonzeronon-lightlike vector fields 119864 119865 on 119872 is defined by theformula

119861 (119864 119865) =

R (119864 119869119864 119865 119869119865)

119892 (119864 119864) 119892 (119865 119865) (93)

For a nonzero non-lightlike vector field 119864 the vector field119869119864 is also non-lightlike and 119864 119869119864 span the 2-dimensionalplane Then the paraholomorphic sectional curvature 119867(119864)is defined as

119867(119864) = 119861 (119864 119864) =

R (119864 119869119864 119864 119869119864)

119892 (119864 119864) 119892 (119864 119864) (94)

The curvature properties of Riemannian submersion andsemi-Riemannian submersion have been extensively studiedin the work of OrsquoNeill [1] and Gray [3]

Let 119861ℎ and 119861V be the paraholomorphic bisectional cur-vatures of horizontal and vertical spaces respectively Let119867ℎ and 119867V be the paraholomorphic sectional curvatures of

horizontal and vertical spaces respectively Let 119861 and 119867 bethe paraholomorphic bisectional and sectional curvatures ofthe base manifold respectively

Proposition 25 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion from an almostpara-Hermitian manifold onto an almost paracontact pseudo-metric manifold and let the fibres of 119891 be pseudo-Riemanniansubmanifolds of 119872 Let 119880 119881 be non-lightlike unit verticalvector fields and let119883 119884 be non-lightlike unit horizontal vectorfields on119872 Then one has

119861 (119880119881) = 119861V(119880 119881) + 119892 (T

119880(119869119881) T

119869119880119881)

minus 119892 (T119869119880(119869119881) T

119880119881)

(95)

Geometry 11

119861 (119883119880) = 119892 ((nabla119880A)119883

119869119883 119869119880) minus 119892 (A119883119869119880A

119869119883119880)

+ 119892 (A119883119880A119869119883119869119880) minus 119892 ((nabla

119869119880A)119883

119869119883119880)

+ 119892 (T119869119880119883T119880(119869119883))

minus 119892 (T119880119883T119869119880(119869119883))

(96)

119861 (119883 119884) = 119861ℎ

(119883 119884) minus 2119892 (A119883(119869119883) A

119884(119869 119884))

+ 119892 (A119869119883119884A119883(119869 119884))

minus 119892 (A119883119884A119869119883(119869 119884))

(97)

Proof Using Definitions (93) and (94) of paraholomorphicsectional curvature and fundamental equations of submer-sion obtained by OrsquoNeill [1] we have (95) (96) and (97)

Corollary 26 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion from an almostpara-Hermitian manifold onto an almost paracontact pseudo-metric manifold If the fibres of 119891 are totally geodesic pseudo-Riemannian submanifolds of119872 then for any non-lightlike unitvertical vector fields 119880 and 119881 one has

119861 (119880119881) = 119861V(119880 119881) (98)

Corollary 27 Let 119891 119872 rarr 119872 be a paracomplex para-contact pseudo-Riemannian submersion from an almost para-Hermitian manifold onto an almost paracontact pseudometricmanifold and let the fibres of 119891 be totally geodesic pseudo-Riemannian submanifolds of119872 If the horizontal distributionis integrable then for any non-lightlike unit horizontal vectorfields119883 and 119884 one has

119861 (119883 119884) = 119861ℎ

(119883 119884) (99)

Proposition 28 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion from an almostpara-Hermitian manifold onto an almost paracontact pseudo-metric manifold and let the fibres of 119891 be pseudo-Riemanniansubmanifolds of119872 Let 119880 and 119883 be non-lightlike unit verticalvector field and non-lightlike unit horizontal vector fieldrespectively Then one has

119867(119880) = 119867V(119880) +

10038171003817100381710038171003817T119880(119869119880)

10038171003817100381710038171003817

2

minus 119892 (T119869119880(119869119880) T

119880119880)

(100)

119867(119883) = 119867 (119883) ∘ 119891 minus 310038171003817100381710038171003817A119883(119869119883)

10038171003817100381710038171003817

2

(101)

Proof The proof is straightforward If we take 119880 = 119881 in (95)and119883 = 119884 in (97) we have (98) and (99)

Corollary 29 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion from an almostpara-Hermitian manifold onto an almost paracontact pseudo-metric manifold If the fibres of 119891 are totally geodesic pseudo-Riemannian submanifolds of 119872 then the total manifold andfibres of119891have the same paraholomorphic sectional curvatures

Proof Since the fibres are totally geodesic T = 0 conse-quently we have

119867(119880) = 119867V(119880) (102)

Corollary 30 Let 119891 119872 rarr 119872 be a paracomplex para-contact pseudo-Riemannian submersion from an almost para-Hermitian manifold onto an almost paracontact pseudometricmanifold and let the fibres of 119891 be totally geodesic pseudo-Riemannian submanifolds of119872 If the horizontal distributionis integrable then the base manifold and horizontal distribu-tion have the same paraholomorphic sectional curvatures

Proof Since the horizontal distribution is integrableA = 0consequently we have

119867(119883) = 119867 (119883) ∘ 119891 (103)

Theorem 31 Let 119891 119872119898

rarr 119872119899 be a paracomplex

paracontact pseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometricmanifold 119872 and let the fibres of 119891 be pseudo-Riemanniansubmanifolds of 119872 If 119880 119881 are the non-lightlike unit verticalvector fields and 119883 119884 are the non-lightlike unit horizontalvector fields then one has

119861 (119880119881) = 119861V(119880 119881) (104)

119861 (119883119880) = minus210038171003817100381710038171003817T11988011988310038171003817100381710038171003817

2

(105)

119861 (119883 119884) = 119861 (119883 119884) ∘ 119891 (106)

Proof Using results of Lemma 19 in (95) we have

119861 (119880119881) = 119861V(119880 119881) minus 119892 (119869 (T

119880119881) 119869 (T

119880119881))

minus 119892 (1198692

(T119880119881) T

119880119881)

= 119861 (119880119881) + 119892 (T119880119881T119880119881) minus 119892 (T

119880119881T119880119881)

= 119861V(119880 119881)

(107)

Applying results of Lemma 19 in (96) we have

119861 (119883119880) = 119892 ((nabla119880A)119883

(119869119883) 119869119880) minus 119892 ((nabla119869119880A)119883

(119869119883) 119880)

+ 21003817100381710038171003817A119883119880

1003817100381710038171003817

2

minus 210038171003817100381710038171003817T11988011988310038171003817100381710038171003817

2

(108)

Since byTheorem 21 the horizontal distribution is integrablewe haveA = 0 which implies

119861 (119883119880) = minus210038171003817100381710038171003817T11988011988310038171003817100381710038171003817

2

(109)

In view ofA = 0 (104) follows from (97)

12 Geometry

Theorem 32 Let 119891 119872119898

rarr 119872119899 be a paracomplex

paracontact pseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometricmanifold 119872 and let the fibres of 119891 be pseudo-Riemanniansubmanifolds of119872 If 119880 119883 are non-lightlike unit vertical andnon-lightlike unit horizontal vector fields respectively then onehas

119867(119880) = 119867V(119880) minus 2

1003817100381710038171003817T1198801198801003817100381710038171003817

2

(110)

119867(119883) = 119867 (119883) ∘ 119891 (111)

Proof Since 119891 is the paracomplex paracontact pseudo-Riemannian submersion from a para-Kahler manifold 119872

onto an almost paracontact pseudometric manifold 119872 by(16) and equations of Lemma 19 we have

119892 (T119869119880(119869119880) T

119880119880) = 119892 (119869

2

(T119880119880) T

119880119880) =

1003817100381710038171003817T1198801198801003817100381710038171003817

2

119892 (T119880(119869119880) T

119880(119869119880)) = minus119892 (T

119880119880T119880119880) = minus

1003817100381710038171003817T1198801198801003817100381710038171003817

2

(112)

and by using the above results in (100) we have

119867(119880) = 119867V(119880) minus

1003817100381710038171003817T1198801198801003817100381710038171003817

2

minus1003817100381710038171003817T119880119880

1003817100381710038171003817

2

= 119867V(119880) minus 2

1003817100381710038171003817T1198801198801003817100381710038171003817

2

(113)

Again since horizontal distribution is integrable we haveA = 0 and putting it in (101) we obtain (111)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Uma Shankar Verma is thankful to University Grant Com-mission New Delhi India for financial support

References

[1] B OrsquoNeill ldquoThe fundamental equations of a submersionrdquo TheMichigan Mathematical Journal vol 13 pp 459ndash469 1966

[2] B OrsquoNeill Semi-Riemannian Geometry with Applications toRelativity vol 103 of Pure and Applied Mathematics AcademicPress New York NY USA 1983

[3] A Gray ldquoPseudo-Riemannian almost product manifolds andsubmersionsrdquo vol 16 pp 715ndash737 1967

[4] J P Bourguignon and H B Lawson A Mathematicians Visitto Kaluza-Klein Theory Rendiconti del Seminario Matematico1989

[5] S Ianus and M Visinescu ldquoSpace-time compactification andRiemannian submersionsrdquo in The Mathematical Heritage GRassias and C F Gauss Eds pp 358ndash371 World ScientificRiver Edge NJ USA 1991

[6] J P Bourguignon and H B Lawson Jr ldquoStability and isolationphenomena for Yang-Mills fieldsrdquo Communications in Mathe-matical Physics vol 79 no 2 pp 189ndash230 1981

[7] B Watson ldquoG 1198661015840

-Riemannian submersions and nonlineargauge field equations of general relativityrdquo in Global AnalysismdashAnalysis on Manifolds T Rassias and M Morse Eds vol 57 ofTeubner-Texte zur Mathematik pp 324ndash349 Teubner LeipzigGermany 1983

[8] C Altafini ldquoRedundant robotic chains on Riemannian submer-sionsrdquo IEEE Transactions on Robotics and Automation vol 20no 2 pp 335ndash340 2004

[9] M T Mustafa ldquoApplications of harmonic morphisms to grav-ityrdquo Journal of Mathematical Physics vol 41 no 10 pp 6918ndash6929 2000

[10] B Watson ldquoAlmost Hermitian submersionsrdquo Journal of Differ-ential Geometry vol 11 no 1 pp 147ndash165 1976

[11] D Chinea ldquoAlmost contact metric submersionsrdquo Rendiconti delCircolo Matematico di Palermo II vol 34 no 1 pp 319ndash3301984

[12] D Chinea ldquoTransference of structures on almost complexcontact metric submersionsrdquo Houston Journal of Mathematicsvol 14 no 1 pp 9ndash22 1988

[13] Y Gunduzalp and B Sahin ldquoPara-contact para-complexpseudo-Riemannian submersionsrdquo Bulletin of the MalaysianMathematical Sciences Society

[14] I Sato ldquoOn a structure similar to the almost contact structurerdquoTensor vol 30 no 3 pp 219ndash224 1976

[15] P K Rashevskij ldquoThe scalar field in a stratified spacerdquoTrudy Seminara po Vektornomu i Tenzornomu Analizu s ikhPrilozheniyami k Geometrii Mekhanike i Fizike vol 6 pp 225ndash248 1948

[16] P Libermann ldquoSur les structures presque paracomplexesrdquoComptes Rendus de lrsquoAcademie des Sciences I vol 234 pp 2517ndash2519 1952

[17] E M Patterson ldquoRiemann extensions which have Kahlermetricsrdquo Proceedings of the Royal Society of Edinburgh AMathematics vol 64 pp 113ndash126 1954

[18] S Sasaki ldquoOn differentiable manifolds with certain structureswhich are closely related to almost contact structure Irdquo TheTohoku Mathematical Journal vol 12 pp 459ndash476 1960

[19] S Zamkovoy ldquoCanonical connections on paracontact mani-foldsrdquo Annals of Global Analysis and Geometry vol 36 no 1pp 37ndash60 2009

[20] D E Blair Riemannian Geometry of Contact and SymplecticManifolds vol 23 of Progress in Mathematics BirkhauserBoston Mass USA 2002

[21] V Cruceanu P Fortuny and P M Gadea ldquoA survey onparacomplex geometryrdquoThe Rocky Mountain Journal of Mathe-matics vol 26 no 1 pp 83ndash115 1996

[22] P M Gadea and J M Masque ldquoClassification of almost para-Hermitian manifoldsrdquo Rendiconti di Matematica e delle sueApplicazioni VII vol 7 no 11 pp 377ndash396 1991

[23] M Falcitelli S Ianus and A M Pastore Riemannian Submer-sions and Related Topics World Scientific River Edge NJ USA2004

[24] E G Rio and D N Kupeli Semi-Riemannian Maps andTheir Applications Kluwer Academic Publisher DordrechtTheNetherlands 1999

[25] B Sahin ldquoSemi-invariant submersions from almost HermitianmanifoldsrdquoCanadianMathematical Bulletin vol 54 no 3 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 3: Research Article Paracomplex Paracontact Pseudo-Riemannian …downloads.hindawi.com/archive/2014/616487.pdf · 2019-07-31 · Research Article Paracomplex Paracontact Pseudo-Riemannian

Geometry 3

An almost paracomplex manifold (119872 119869) is such thatthe two eigenbundles 119879

+

119872 and 119879minus

119872 corresponding torespective eigenvalues +1 and minus1 of 119869 have the same rank[21 22]

An almost para-Hermitianmanifold (119872 119869 119892) is a smoothmanifold endowed with an almost paracomplex structure 119869and a pseudo-Riemannian metric 119892 such that

119892 (119869119883 119869119884) = minus119892 (119883 119884) forall119883 119884 isin Γ (119879119872) (16)

Here the metric 119892 is neutral that is 119892 has signature (119898119898)The fundamental 2-form of the almost para-Hermitian

manifold is defined by

119865 (119883 119884) = 119892 (119883 119869119884) (17)

We have the following properties [21 22]

119892 (119869119883 119884) = minus119892 (119883 119869119884) (18)

119865 (119883 119884) = minus119865 (119884119883) (19)

119865 (119869119883 119869119884) = minus119865 (119883 119884) (20)

3119889119865 (119883 119884 119885)

= 119883 (119865 (119884 119885)) minus 119884 (119865 (119883 119885)) + 119885 (119865 (119883 119884))

minus 119865 ([119883 119884] 119885) + 119865 ([119883 119885] 119884) minus 119865 ([119884 119885] 119883)

(21)

(nabla119883119865) (119884 119885) = 119892 (119884 (nabla

119883119869) 119885) = minus119892 (119885 (nabla

119883119869) 119884) (22)

3119889119865 (119883 119884 119885) = (nabla119883119865) (119884 119885) + (nabla

119884119865) (119885119883)

+ (nabla119885119865) (119883 119884)

(23)

the co-differential (120575119865) (119883) =2119898

sum

119894=1

120576119894(nabla119890119894119865) (119890119894 119883) (24)

An almost para-Hermitian manifold is called

(i) para-Hermitian if 119873119869= 0 equivalently (nabla

119869119883119869)119869119884 +

(nabla119883119869)119884 = 0

(ii) para-Kahler if for any 119883 isin Γ(119879119872) nabla119883119869 = 0 that is

nabla119869 = 0(iii) almost para-Kahler if 119889119865 = 0(iv) nearly para-Kahler if (nabla

119883119869)119883 = 0

(v) almost semi-para-Kahler if 120575119865 = 0(vi) semi-para-Kahler if 120575119865 = 0 and119873

119869= 0

23 Pseudo-Riemannian Submersion Let (119872119898

119892) and(119872119899

119892) be two connected pseudo-Riemannian manifolds ofindices 119904 (0 le 119904 le 119898) and 119904 (0 le 119904 le 119899) respectively with119904 ge 119904

A pseudo-Riemannian submersion is a smooth map 119891

119872119898

rarr 119872119899 which is onto and satisfies the following

conditions [2 3 23 24]

(i) The derivative map 119891lowast119901

119879119901119872 rarr 119879

119891(119901)119872 is

surjective at each point 119901 isin 119872

(ii) The fibres 119891minus1(119902) of 119891 over 119902 isin 119872 are eitherpseudo-Riemannian submanifolds of 119872 ofdimension (119898 minus 119899) and index ] or the degeneratesubmanifolds of 119872 of dimension (119898 minus 119899) andindex ] with degenerate metric 119892

|119891minus1(119902)

of type(0 0 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

120583-times minus minus minus minus⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

]-times + + + +⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

(119898minus119899minus120583minus])-times)) where

120583 = dim(V119901capH119901) and ] = 119904 minus 119904 = index of 119892

|119891minus1(119902)

(iii) 119891lowastpreserves the length of horizontal vectors

We denote the vertical and horizontal projections of avector field 119864 on 119872 by 119864V (or by V119864) and 119864

ℎ (or by ℎ119864)respectively A horizontal vector field 119883 on 119872 is said to bebasic if 119883 is 119891-related to a vector field 119883 on119872 Thus everyvector field119883 on119872 has a unique horizontal lift119883 on119872

Lemma 1 (see [1 23]) If 119891 119872 rarr 119872 is a pseudo-Riemannian submersion and119883 119884 are basic vector fields on119872that are 119891-related to the vector fields 119883 119884 on119872 respectivelythen one has the following properties

(i) 119892(119883 119884) = 119892(119883 119884) ∘ 119891(ii) ℎ[119883 119884] is a vector field and ℎ[119883 119884] = [119883 119884] ∘ 119891(iii) ℎ(nabla

119883119884) is a basic vector field 119891-related to nabla

119883119884 where

nabla and nabla are the Levi-Civita connections on119872 and119872respectively

(iv) [119864 119880] isin V for any vector field 119880 isin V and for anyvector field 119864 isin Γ(119879119872)

A pseudo-Riemannian submersion 119891 119872 rarr 119872

determines tensor fieldsT andA of type (1 2) on119872 definedby formulas [1 2 23]

T (119864 119865) = T119864119865 = ℎ (nablaV119864V119865) + V (nablaV119864ℎ119865) (25)

A (119864 119865) = A119864119865 = V (nabla

ℎ119864ℎ119865) + ℎ (nabla

ℎ119864V119865)

for any 119864 119865 isin Γ (119879119872)

(26)

Let 119883 119884 be horizontal vector fields and let 119880 119881 bevertical vector fields on119872 Then one has

T119880119883 = V (nabla

119880119883) T

119880119881 = ℎ (nabla

119880119881) (27)

nabla119880119883 = T

119880119883 + ℎ (nabla

119880119883) (28)

T119883119865 = 0 T

119864119865 = TV119864119865 (29)

nabla119880119881 = T

119880119881 + V (nabla

119880119881) (30)

A119883119884 = V (nabla

119883119884) A

119883119880 = ℎ (nabla

119883119880) (31)

nabla119883119880 = A

119883119880 + V (nabla

119883119880) (32)

A119880119865 = 0 A

119864119865 = A

ℎ119864119865 (33)

nabla119883119884 = A

119883119884 + ℎ (nabla

119883119884) (34)

4 Geometry

ℎ (nabla119880119883) = ℎ (nabla

119883119880) = A

119883119880 (35)

A119883119884 =

1

2V [119883 119884] (36)

A119883119884 = minusA

119884119883 (37)

T119880119881 = T

119881119880 (38)

for all 119864 119865 isin Γ(119879119872)MoreoverT

119880119881 coincideswith second fundamental form

of the submersion of the fibre submanifoldsThe distributionH is completely integrable In view of (37) and (38) A isalternating on the horizontal distribution andT is symmetricon the vertical distribution

3 Paracomplex Paracontact Pseudo-Riemannian Submersions

In this section we introduce the notion of pseudo-Riemannian submersion from almost paracomplex mani-folds onto almost paracontact pseudometric manifolds illus-trate examples and study the transference of structures ontotal manifolds and base manifolds

Definition 2 Let (1198722119898 119869 119892) be an almost para-Hermitianmanifold and let (1198722119899+1 120601 120585 120578 119892) be an almost paracontactpseudometric manifold

A pseudo-Riemannian submersion 119891 119872 rarr 119872 is calledparacomplex paracontact pseudo-Riemannian submersion ifthere exists a 1-form 120578 on119872 such that

119891lowast∘ 119869 = 120601 ∘ 119891

lowast+ 120578 otimes 120585 (39)

Since for each 119901 isin 119872119891lowast119901

is a linear isometry betweenhorizontal spacesH

119901and tangent spaces119879

119891(119901)119872 there exists

an induced almost paracontact structure (120601ℎ

120578ℎ

120585ℎ

119892) on(2119899 + 1)-dimensional horizontal distribution H such that120601ℎ

|

Dℎbehave just like the fundamental collineation of almost

paracomplex structure 119869 on ker 120578ℎ = Dℎ

and 120601ℎ

Dℎ

rarr Dℎ

is an endomorphism such that 120601ℎ

= 119869|ker 120578ℎ

and the rank of

120601ℎ

= 2119899 where dim(Dℎ

) = 2119899It follows that for any 119883ℎ isin D

ℎ

120578ℎ(119883ℎ) = 0 whichimplies that 1198692

|

Dℎ(119883ℎ

) = (120601ℎ

)2

(119883ℎ

) = 119883ℎ for any 119883ℎ isin D

ℎ

andH = Dℎ

oplus 120585ℎ

[18]

Definition 3 (see [25]) A pseudo-Riemannian submersion119891 119872 rarr 119872 is called semi-119869-invariant submersion if thereis a distributionD

1sube ker119891

lowastsuch that

ker119891lowast= D1oplusD2 (40)

119869 (D1) = D

1 119869 (D

2) sube (ker119891

lowast)perp

(41)

whereD2is orthogonal complementary toD

1in ker119891

lowast

Proposition 4 Let119891 1198722119898 rarr 1198722119899+1 be a paracomplex par-

acontact pseudo-Riemannian submersion and let the fibres of119891 be pseudo-Riemannian submanifolds of119872 Then the fibres119891minus1

(119902) 119902 isin 119872 are semi-119869-invariant submanifolds of 119872 ofdimension (2119898 minus 2119899 minus 1)

Proof Let 119880 isinV Then

119891lowast(119869119880) = 120601 (119891

lowast(119880)) + 120578 (119880) 120585

997904rArr 119891lowast119869 (119880) minus 120578 (119880) 120585

ℎ

= 0

(42)

where 119891lowast120585ℎ

= 120585Thus we have

119869 (119880) minus 120578 (119880) 120585ℎ

= 120601 (119880) for some 120601 (119880) isinV (43)

By (19) we get 119892(120585ℎ

119869(120585ℎ

)) = 0 = 119892(120585 119891lowast(119869(120585ℎ

))) = 0As 119892 is nondegenerate on119872 we have

119891lowast(119869 (120585ℎ

)) = 0 that is 119869 (120585ℎ

) isinV (44)

Taking 119880 = 119869120585ℎ

in (43) we obtain

120585ℎ

minus 120578 (119869120585ℎ

) 120585ℎ

= 120601(119869120585ℎ

) (45)

Since fibre 119891minus1(119902) is an odd dimensional submanifold thereexists an associated 1-form 120578

V which is restriction of 120578 on fibresubmanifold 119891minus1(119902) 119902 isin 119872 and a characteristic vector field120585V= 119869120585ℎ

such that 120601(120585V) = 0 So we have 120578V(120585

V) = 1

Let us put ker 120578V = D1andD

2= 120585

V

Then ker119891lowast= D1oplus D2and 119869(D

1) = D

1 119869(D

2) =

119869120585V = 120585

ℎ

sube (ker119891lowast)perp

Hence the fibres 119891minus1(119902) are semi-119869-invariant submani-folds of119872

Corollary 5 Let 119891 1198722119898

rarr 1198722119899+1 be a paracomplex

paracontact pseudo-Riemannian submersion and let the fibresof 119891 be pseudo-Riemannian submanifolds of 119872 Then thefibres 119891

minus1

(119902) are almost paracontact pseudometric mani-folds with almost paracontact pseudo-Riemannian structures(120601

V 120585

V 120578

V 119892

V) 119902 isin 119872 where 120585

V= 119869(120585

ℎ

) 120578V = 120578|V and

119892V= 119892

Proof Since 119891minus1(119902) are semi-119869-invariant submanifolds of119872of odd dimension 2119903 + 1 = 2119898 minus 2119899 minus 1 (39) implies

119869 (119880) = 120601V119880 + 120578

V(119880) 120585ℎ

(46)

for any 119880 isinV

Geometry 5

On operating 119869 on both sides of the above equation weget

119880 = 120601V(120601

V(119880)) + 120578

V(120601

V(119880)) 120585

ℎ

+ 120578V(119880) 120585

V (47)

where 119869(120585ℎ

) = 120585V

Equating horizontal and vertical components we have

119880 = 120601V(120601

V(119880)) + 120578

V(119880) 120585

V 120578

V∘ 120601

V(119880) = 0

997904rArr (120601V)2

(119880) = 119880 minus 120578V(119880) 120585

V 120578

V∘ 120601

V= 0

120601V(120585

V) = 0 120578

V(120585

V) = 1

(48)

Hence (120601V 120585

V 120578

V 119892

V) is almost paracontact pseudometric

structure on the fibre 119891minus1(119902) 119902 isin 119872

Proposition 6 Let 119891 1198722119898

rarr 1198722119899+1 be a paracomplex

paracontact pseudo-Riemannian submersion and let the fibresof 119891 be pseudo-Riemannian submanifolds of 119872 Let 119883 119884 bebasic vector fields 119891-related to 119883 119884 respectively Let 120578 and 120578be 1-forms on the total manifold119872 and the base manifold119872respectively Then one has the following

(i) The characteristic vector field 119869120585ℎ

is a vertical vectorfield

(ii) 119891lowastlowast120578 = 120578ℎ where 119891lowast

lowast120578 is pullback of 120578 through 119891

lowast

(iii) 120578ℎ(119880) = 0 for any vertical vector field 119880(iv) 120578V(119883) = 0 for any horizontal vector field119883

Remark 7 Results (ii) and (iv) are analogue version of results(i) and (iii) of Proposition 4 of [13]

Proof (i) By Corollary 5 (120601V 120585

V 120578

V 119892

V) is almost paracontact

pseudometric structure on 119891minus1(119902) We have

0 = 119892 (120585ℎ

119869120585ℎ

) = 119892(119891lowast(120585ℎ

) 119891lowast(119869120585ℎ

))

= 119892(120585 119891lowast(119869120585ℎ

))

(49)

Now

119891lowast(119869120585ℎ

) = 120601 ∘ 119891lowast120585ℎ

+ 120578 (120585ℎ

) 120585 = 120578 (120585ℎ

) 120585 (50)

so we have

0 = 119892(120585 120578 (120585ℎ

) 120585) = 120578 (120585ℎ

)119892 (120585 120585) = 120578 (120585ℎ

) (51)

Thus 119891lowast(119869120585ℎ

) = 0Hence 119869120585

ℎ

is a vertical vector field(ii) Since 119891 119872

2119898

rarr 1198722119899+1 is smooth submersion

120578ℎ

= 120578|H

is restriction of 120578 on the horizontal distribution

H and 119891lowast119901

H119901rarr 119879119891(119901)

119872 is a linear isometry for any119883119901isinH119901 we get

120578ℎ

119901(119883119901) = 120576119892

119901(120585ℎ

119901 119883119901) = 119892

119891(119901)(119891lowast119901120585ℎ

119901 119891lowast119901119883119901)

= 119892119891(119901)

(120585119891(119901)

119883119891(119901)

) = 120578119891(119901)

(119883119891(119901)

) = 119891lowast

lowast120578119901(119883119901)

(52)

Hence pullback 119891lowastlowast120578 = 120578ℎ

Results (iii) and (iv) immediately follow from the previ-ous results

Example 8 Let (R42 119869 119892) be a paracomplex pseudometric

manifold and let (R31 120601 120585 120578 119892) be an almost paracontact

pseudometric manifoldDefine a submersion 119891 R4

2 (1199091 1199092 1199101 1199102)119905

rarr

R31 (119906 V 119908)119905 by

119891 ((1199091 1199092 1199101 1199102)119905

) 997891997888rarr (1199091+ 1199092+ 31199101+ 21199102

31199091+ 21199092+ 1199101+ 1199102

51199091+ 31199092+ 51199101+ 31199102)119905

(53)

Then the kernel of 119891lowastis

V = ker119891lowast= Span119881

1=

120597

1205971199091

minus 2120597

1205971199092

minus120597

1205971199101

+ 2120597

1205971199102

(54)

which is the vertical distribution admitting one lightlikevector field that is fibre is degenerate submanifold of R4

2

The horizontal distribution is

H = (ker119891lowast)perp

= Span1198831=

120597

1205971199091

minus120597

1205971199101

1198832=

120597

1205971199092

+ 2120597

1205971199101

1198833= 2

120597

1205971199101

+120597

1205971199102

(55)

For any real 119896 the horizontal characteristic vector field 120585ℎ

isgiven by

120585ℎ

= 119896120597

1205971199091

minus (2119896 minus1

3)

120597

1205971199092

minus (119896 minus 1)120597

1205971199101

+ (2119896 minus5

3)

120597

1205971199102

(56)

which is 119891-related to the characteristic vector field 120585 = 120597120597119908Moreover there exists one form 120578 = 5119889119909

1+3119889119909

2+51198891199101+

31198891199102on (R4

2 119869 119892) such that the submersion satisfies (39)

Example 9 Let (R63 119869 119892) be an almost paracomplex pseudo-

Riemannian manifold and let (R31 120601 120585 120578 119892) be an almost

6 Geometry

paracontact pseudo-Riemannian manifold Consider a sub-mersion 119891 R6

3 (1199091 1199092 1199093 1199101 1199102 1199103)119905

rarr R31 (119906 V 119908)119905

defined by

119891 ((1199091 1199092 1199093 1199101 1199102 1199103)119905

)

997891997888rarr (1199091+ 1199092

radic2

1199101+ 1199102

radic2

1199102+ 1199103

radic2

)

119905

(57)

Then there exists one form 120578 = (1198891199092+ 1198891199093)radic2 on (R6

3 119869 119892)

such that (39) is satisfied The kernel of 119891lowastis

V = ker119891lowast

= Span1198811=

120597

1205971199091

minus120597

1205971199092

1198812=

120597

1205971199101

minus120597

1205971199102

+120597

1205971199103

1198813=

120597

1205971199093

(58)

which is vertical distribution admitting non-lightlike vectorfields that is the fibre is nondegenerate submanifold of(R63 119869 119892)The horizontal distribution is

H = Span1198831=

120597

1205971199091

+120597

1205971199092

1198832= minus

120597

1205971199101

+120597

1205971199103

1198833=

120597

1205971199101

+120597

1205971199102

(59)

Example 10 Let (R42 119869 119892) be a paracomplex pseudometric

manifold and let (R31 120601 120585 120578 119892) be an almost paracontact

pseudometric manifoldConsider a submersion 119891 R4

2 (1199091 1199092 1199101 1199102)119905

rarr R31

(119906 V 119908)119905 defined by

119891 ((1199091 1199092 1199101 1199102)119905

) 997891997888rarr (1199091 1199101 1199102)119905

(60)

Then the kernel of 119891lowastis

V = ker119891lowast= Span119881

1=

120597

1205971199092

(61)

which is the vertical distribution and the restriction of 119892 tothe fibres of 119891 is nondegenerate

The horizontal distribution is

H = (ker119891lowast)perp

= Span119883 =120597

1205971199091

119884 =120597

1205971199101

120585ℎ

=120597

1205971199102

(62)

The characteristic vector field 120585 = 120597120597119908 on R31has unique

horizontal lift 120585ℎ

which is the characteristic vector field onhorizontal distributionH of R4

2

We also have

119892 (119883119883) = 119892 (119891lowast119883119891lowast119883) = minus1

119892 (119884 119884) = 119892 (119891lowast119884 119891lowast119884) = 1

119892 (120585ℎ

120585ℎ

) = 119892(119891lowast120585ℎ

119891lowast120585ℎ

) = 119892 (120585 120585) = 1

(63)

Thus the smooth map 119891 is a pseudo-Riemannian submer-sion

Moreover we obtain that there exists a 1-form 120578 = 1198891199092on

R42such that 120578(119869120585

ℎ

) = 1 120578(120585ℎ

) = 0 and the map 119891 satisfies

119891lowast119869119883 = 120601119891

lowast119883 + 120578 (119883) 120585

119891lowast119869119884 = 120601119891

lowast119884 + 120578 (119884) 120585

119891lowast119869120585ℎ

= 120601119891lowast120585ℎ

+ 120578 (120585ℎ

) 120585

(64)

Hence the map 119891 is a paracomplex paracontact pseudo-Ri-emannian submersion from R4

2on to R3

1

Proposition 11 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion and let the fibresof 119891 be pseudo-Riemannian submanifolds of 119872 Let 119883 119884 bebasic vector fields 119891-related to119883 119884 respectively Then 119869(119883) minus120576119892(119883 120585

ℎ

)120585ℎ

is 119891-related to 120601119883

Proof Since119883 is 119891-related to vector field119883 on119872 we have

120578 (119883) = 120578V+ 120578ℎ

(119883) = 0 + 120578ℎ

(119883) = 120576119892 (119883 120585ℎ

)

997904rArr 119891lowast(119869119883) = 120601119883 + 120578

ℎ

(119883) 120585

997904rArr 119891lowast119869119883 minus 120576119892 (119883 120585

ℎ

) 120585ℎ

= 120601119883

(65)

Hence 119869(119883) minus 120576119892(119883 120585ℎ

)120585ℎ

is 119891-related to 120601119883

Proposition 12 Let 119891 119872 rarr 119872 be a paracomplex para-contact pseudo-Riemannian submersion and let the fibres of 119891be pseudo-Riemannian submanifolds of 119872 Let V and H bethe vertical and horizontal distributions respectively If 120585

ℎ

isthe basic characteristic vector field of horizontal distribution119891-related to the characteristic vector field 120585 of base manifoldthen

(i) 119869V sub DVoplus 119869120585ℎ

oplus 120585ℎ

(ii) 119869H sub Dℎ

oplus 120585ℎ

oplus 119869120585ℎ

Proof (i) Let 119880 isin V Then 119880 = 119886119880|DV + 119887119869120585

ℎ

for 119886 119887 isin

119862infin

(119872) as 119869120585ℎ

= 120585Vis characteristic vector field on odd

Geometry 7

dimensional fibre submanifold 119891minus1(119902) of119872 119902 isin 119872 We get

119869119880 = 119886119869119880|DV + 119887119869

2

120585ℎ

= 119886119869119880|DV + 119887120585

ℎ

isinV oplus 120585ℎ

997904rArr 119869V subV oplus 120585ℎ

(66)

Again let 119881 isinV oplus 120585ℎ

Then 119881 = 119886119881|DV + 119887119869120585

ℎ

+ 119888120585ℎ

where

120578V(119881|DV ) = 0 D

V= ker 120578V 119886119881

|DV + 119887119869120585

ℎ

isin V and 119886 119887 119888 isin119862infin

(119872) We have

119869119881 = 119886119869119881|DV + 119887120585

ℎ

+ 119888119869120585ℎ

= (119886119869119881|DV + 119888119869120585

ℎ

)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

isinV

+ 119887120585ℎ

⏟⏟⏟⏟⏟⏟⏟

isin120585

ℎ

isinV oplus 120585ℎ

(67)

Now by (39) we get

119891lowast119869119881 = 120601 (119891

lowast119881) + 120578 (119891

lowast(119881)) 120585

= 119888 120601 (119891lowast120585ℎ

) + 120578 (120585) 120585

= 119888120585 isin 120585 sube 119869V

(68)

We get 119869119881 notinVHence 119869V subV oplus 120585

ℎ

that is 119869V sub DVoplus 119869120585ℎ

oplus 120585ℎ

(ii) Let 119883 = 119886119883

|

Dℎ+ 119887120585ℎ

isin H where H = Dℎ

oplus

120585ℎ

ker 120578ℎ = Dℎ

and 119886 119887 isin 119862infin(119872) Then

119869119883 = 119886119869119883|

Dℎ+ 119887119869120585ℎ

isinH oplus 119869120585ℎ

(69)

which implies that 119869H subH oplus 119869120585ℎ

Again let119884 isinHoplus119869120585

ℎ

Then119884 = 119886119884|

Dℎ+119887120585ℎ

+119888119869120585ℎ

notinHfor 119886 119887 119888 isin 119862infin(119872) We have

119869 119884 = 119886119869119884|

Dℎ+ 119887119869120585ℎ

+ 1198881198692

120585ℎ

= 119886119869119884|

Dℎ+ 119887119869120585ℎ

+ 119888120585ℎ

= 119885 + 119887119869120585ℎ

isinH oplus 119869120585ℎ

for some 119885 = 119886119869119884|

Dℎ+ 119888120585ℎ

isinH

(70)

We obtain 119869 119884 notinHHence 119869H sub H oplus 119869(120585

ℎ

) that is 119869H sub Dℎ

oplus 120585ℎ

oplus

119869120585ℎ

Example 13 Let (R63 119869 119892) be an almost paracomplex pseudo-

Riemannian manifold and let (R31 120601 120585 120578 119892) be an almost

paracontact pseudo-Riemannian manifold Consider a sub-mersion 119891 R6

3 (1199091 1199092 1199093 1199101 1199102 1199103)119905

rarr R31 (119906 V 119908)119905

defined by

119891 ((1199091 1199092 1199093 1199101 1199102 1199103)119905

) 997891997888rarr (1199091+ 1199092

radic2

1199101+ 1199102

radic2

1199103)

119905

(71)

Then the kernel of 119891lowastis

V = ker119891lowast

= Span1198811=

120597

1205971199091

minus120597

1205971199092

1198812=

120597

1205971199101

minus120597

1205971199102

120585V=

120597

1205971199093

(72)

which is the vertical distribution and the restriction of 119892 tothe fibres of 119891 is nondegenerate

The horizontal distribution is

H = (ker119891lowast)perp

= Span1198831=

120597

1205971199091

+120597

1205971199092

1198832=

120597

1205971199101

+120597

1205971199102

120585ℎ

=120597

1205971199103

(73)

The characteristic vector field 120585 = 120597120597119908 on R31has unique

horizontal lift 120585ℎ

which is the characteristic vector field onthe horizontal distributionH of R6

3

We also have

119892 (1198831 1198831) = 119892 (119891

lowast1198831 119891lowast1198831) = minus2

119892 (1198832 1198832) = 119892 (119891

lowast1198832 119891lowast1198832) = 2

119892 (120585ℎ

120585ℎ

) = 119892(119891lowast120585ℎ

119891lowast120585ℎ

) = 119892 (120585 120585) = 1

(74)

Thus the smooth map 119891 is a pseudo-Riemannian submer-sion

Also we obtain that there exists a 1-form 120578 = 1198891199093on R63

such that 120578(119869120585ℎ

) = 1 120578(120585ℎ

) = 0 and the map 119891 satisfies

119891lowast1198691198831= 120601119891lowast1198831+ 120578 (119883

1) 120585

119891lowast1198691198832= 120601119891lowast1198832+ 120578 (119883

2) 120585

119891lowast119869120585ℎ

= 120601119891lowast120585ℎ

+ 120578 (120585ℎ

) 120585

(75)

Hence the map 119891 is a paracomplex paracontact pseudo-Riemannian submersion from R6

3onto R3

1

Moreover we observe that for this submersion119891 we have

119869V subV oplus 120585ℎ

119869H subH oplus 119869120585ℎ

(76)

which verifies Proposition 12

8 Geometry

Proposition 14 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion and let the fibresof 119891 be pseudo-Riemannian submanifolds of 119872 Let 119883 119884 bebasic vector fields 119891-related to 119883 119884 respectively Let 119865 and Φbe the second fundamental forms and let nabla and nabla be the Levi-Civita connection on the total manifold119872 and base manifold119872 respectively Then one has

(i) 119891lowast((nabla119883119869)119884) = (nabla

119883120601)119884 + 120576119892(119884 nabla

119883120585)120585 + 120578(119884)nabla

119883120585

(ii) 119865 = 119891lowastlowastΦ + 120576120578 otimes 120578

(iii) 119891lowast((nabla119883119865)(119884 119885)) = (nabla

119883Φ)(119884 119885) + 120578(119884)119892(119885 nabla

119883120585) +

120578(119885)119892(119884 nabla119883120585)

Proof (i) In view of Definition 2 and Proposition 11 we have

119891lowast((nabla119883119869) 119884) = 119891

lowast(nabla119883(119869 119884) minus 119869 (nabla

119883119884))

= nabla119883(119891lowast(119869 119884)) minus 119891

lowast(119869 (nabla119883119884))

= nabla119883(120601119884) + nabla

119883(120578 (119884) 120585) minus 120601 (nabla

119883119884)

minus 120578 (nabla119883119884) 120585

= (nabla119883120601)119884 + nabla

119883(120576119892 (119884 120585) 120585) minus 120578 (nabla

119883119884) 120585

= (nabla119883120601)119884 + 120576119892 (nabla

119883119884 120585) 120585 + 120576119892 (119884 nabla

119883120585) 120585

+ 120576119892 (119884 120585) nabla119883120585 minus 120576119892 (nabla

119883119884 120585) 120585

= (nabla119883120601)119884 + 120576119892 (119884 nabla

119883120585) 120585 + 120578 (119884) nabla

119883120585

(77)

(ii) Since119891lowastlowastΦ is pullback ofΦ through the linearmap119891

lowast

we get

119891lowast

lowastΦ(119883 119884) = Φ (119883 119884) ∘ 119891 = 119892 (119883 120601119884) ∘ 119891

= 119892 (119883 119869 119884) minus 120576120578 (119884) 120578 (119883)

= 119865 (119883 119884) minus 120576120578 (119883) 120578 (119884)

(78)

which implies 119865 = 119891lowastlowastΦ + 120576120578 otimes 120578

(iii) By (23) we have

119891lowast((nabla119883119865) (119884 119885)) = 119892 (119891

lowast(119884) 119891

lowast((nabla119883119869)119885)) (79)

Now using (i) in the above equation we get (iii)

Theorem 15 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion and let the fibresof 119891 be pseudo-Riemannian submanifolds of 119872 Let 119883 119884 bebasic vector fields 119891-related to 119883 119884 respectively If the totalspace is para-Hermitian manifold then the almost paracontactstructure of base space is normal

Moreover if the almost paracontact structure of base spaceis normal then the Nijenhuis tensor of total space is vertical

Proof The Nijenhuis tensors 119873119869and 119873

120601of almost para-

complex structure 119869 and almost paracontact structure 120601 arerespectively defined by (8) and (11)

Using Definition 2 and properties of Sections 21 and 22we get the following identity

119891lowast(119873119869(119883 119884)) = 119873

(1)

(119883 119884) + 2119889120578 (120601119883 119884) 120585

minus 2119889120578 (120601119884119883) 120585

+ 2120578 (119883) 119889120578 (120585 119884) 120585 minus 2120578 (119884) 119889120578 (120585 119883) 120585

minus 120578 (119884)119873(3)

(119883) + 120578 (119883)119873(3)

(119884)

(80)

Using (12) (13) (14) and (15) (80) reduces to

119891lowast(119873119869(119883 119884)) = 119873

(1)

(119883 119884) + 119873(2)

(119883 119884) 120585

+ 120578 (119883)119873(4)

(119884) 120585

minus 120578 (119884)119873(4)

(119883) 120585 minus 120578 (119884)119873(3)

(119883)

+ 120578 (119883)119873(3)

(119884)

(81)

Since 119873119869(119883 119884) = 0 it follows from (81) that

tensors 119873(1) 119873(2) 119873(3) and 119873(4) vanish togetherHence the almost paracontact structure of base space is

normalConversely let the almost paracontact structure of the

base space be normalThen (81) implies that 119891

lowast(119873119869(119883 119884)) = 0

Hence119873119869(119883 119884) is vertical

Corollary 16 Let119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion and let the fibres of 119891 bepseudo-Riemannian submanifolds of 119872 Let 119883 119884 be basicvector fields 119891-related to 119883 119884 respectively Let the total spacebe para-Hermitian manifold and119873(1) vanishes Then the basespace is paracontact pseudometric manifold if and only if 120585 iskilling

Proof Let the total space be para-Hermitian and 119873(1) van-ishes Then from (80) we have

0 = 2119889120578 (120601119883 119884) 120585 minus 2119889120578 (120601119884119883) 120585 + 2120578 (119883) 119889120578 (120585 119884) 120585

minus 2120578 (119884) 119889120578 (120585 119883) 120585 minus 120578 (119884) (L120585120601)119883 + 120578 (119883) (L

120585120601)119884

(82)

If 120585 is killing then we have L120585120601 = 0 It immediately follows

from (82) that

119889120578 (120601119883 119884) minus 120578 (120601119884119883) + 120578 (119883) 119889120578 (120585 119884)

minus 120578 (119884) 119889120578 (120585 119883) = 0

(83)

In view of (6) and (7) the above equation gives 119889120578 = ΦConversely let the base space be paracontact Then 119889120578 =

ΦUsing (6) (7) and (82) we getL

120585120601 = 0

Hence the characteristic vector field 120585 is killing

Geometry 9

Theorem 17 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion and let the fibres of 119891 bepseudo-Riemannian submanifolds of 119872 Let 119883 119884 be basicvector fields 119891-related to 119883 119884 respectively If the total spaceis para-Kahler then the base space is paracosymplectic Theconverse is true if nabla

119883119869 is vertical

Proof We have for any 119883119884 isin Γ(119879119872) (nabla119883120601)119884 = 0 which

gives 119892(119885 (nabla119883120601)119884) = 0 for any 119885 isin Γ(119879119872)

From Proposition 14 we have

119892 (119885 (nabla119883120601)119884) + 120576120578 (119884) (nabla

119883120578)119885 + 120576120578 (119885) (nabla

119883120578) 119884 ∘ 119891

= 119892 (119885 (nabla119883119869) 119884)

(84)

Let nabla 119869 = 0 that is the total space is para-KahlerThen from(84) we obtain nabla120601 = 0 and nabla120578 = 0 Hence the base space isparacosymplectic

Again let (nabla119883120601)119884 = 0 and nabla120578 = 0 Then 119892(119885 (nabla

119883119869)119884) =

0 which implies that (nabla119883119869)119884 is a vertical vector field

Theorem 18 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion and let the fibres of 119891 bepseudo-Riemannian submanifolds of 119872 Let 119883 119884 and 119885 bebasic vector fields 119891-related to 119883 119884 and 119885 respectively If120578(119883)119885(120578(119884))+120578(119884)119883(120578(119885))minus120578(119883)119884(120578(119885)) = 0 then the totalspace is almost para-Kahler if and only if the base space119872 isan almost paracosymplectic manifold

Proof We have the following equation

3119889119865 (119883 119884 119885)

= 3 (119891lowast

lowast119889Φ) (119883 119884 119885) + 2120576120578 (119885) 119889120578 (119883 119884)

minus 2120576120578 (119884) 119889120578 (119883 119885)

+ 2120576120578 (119883) 119889120578 (119884 119885) + 2120576120578 (119883)119885 (120578 (119884))

+ 2120576120578 (119884)119883 (120578 (119885))

minus 2120576120578 (119883)119884 (120578 (119885))

(85)

If 119889120578 = 0 119889Φ = 0 and 120578(119883)119885(120578(119884)) + 120578(119884)119883(120578(119885)) minus

120578(119883)119884(120578(119885)) = 0 then from (85) we have 119889119865 = 0 Hencethe total space is almost para-Kahler

Conversely let 119889119865 = 0 and 120578(119883)119885(120578(119884))+120578(119884)119883(120578(119885))minus120578(119883)119884(120578(119885)) = 0

By using the above equation in (85) we have 119889120578 = 0 and119889Φ = 0

Hence the base space is almost paracosymplectic

Nowwe investigate the properties of fundamental tensorsT andA of a pseudo-Riemannian submersion

Lemma 19 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometric manifold119872 and

let the fibres of 119891 be pseudo-Riemannian submanifolds of 119872Then for any horizontal vector fields119883 119884 and for any verticalvector fields 119880 119881 on119872 one has

(i) A119883(119869 119884) = 119869(A

119883119884)

(ii) A119869119883(119884) = 119869(A

119883119884)

(iii) T119880(119869 119881) = 119869(T

119880119881)

(iv) T119869119880119881 = 119869(T

119880119881)

Proof The proof follows using similar steps as in Lemmas 3and 4 of [13] so we omit it

Lemma 20 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometric manifold119872 andlet the fibres of 119891 be pseudo-Riemannian submanifolds of 119872Then for any vector fields 119864 119865 on119872 one has

(i) A119864(119869119865) = 119869(A

119864119865)

(ii) T119864(119869119865) = 119869(T

119864119865)

Proof The proof follows from (37) and (38)

Theorem 21 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometric manifold119872 andlet the fibres of 119891 be pseudo-Riemannian submanifolds of 119872Then the horizontal distribution is integrable

Proof For any vertical vector field 119880 we have

119892 (119869 (A119883119884) 119880) = 119892 (A

119883119869 119884119880)

= minus119892 (119869119884A119883119880)

= minus119892 (119869119884 ℎ (nabla119880119883))

= 119892 (119884 ℎ (119869 (nabla119880119883)))

= 119892 (119884 ℎ (minusnabla119880119869)119883

+nabla119880(119869119883))

= 119892 (119884 ℎ nabla119880(119869119883)) = 119892 (119884A

119869119883119880)

= minus119892 (A119869119883119884119880) = minus119892 (119869 (A

119883119884) 119880)

(86)

Thus 119892(119869(A119883119884) 119880) = 0 which is true for all119883 and 119884

SoA119883119884 = 0

Hence the horizontal distribution is integrable

Theorem 22 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometric manifold119872 andlet the fibres of 119891 be pseudo-Riemannian submanifolds of 119872Then the submersion is an affine map onH

10 Geometry

Proof The second fundamental form of 119891 is defined by

(nabla119891lowast) (119864 119865) = (nabla

119891

119864119891lowast(119865)) ∘ 119891 minus 119891

lowast(nabla119864119865) (87)

where 119864 119865 isin Γ(119879119872) and nabla119891 is pullback connection of Levi-Civita connection nabla on119872 with respect to 119891

We have for any119883119884 isinH

(nabla119891lowast) (119883 119884) = (nabla

119891

119883

119891lowast(119884)) ∘ 119891 minus 119891

lowast(nabla119883119884) (88)

By using Lemma 1 we have 119891lowast(ℎ(nabla119883119884)) = (nabla

119883119884) ∘ 119891 which

implies nabla119891lowast= 0

Hence the submersion 119891 is an affine map onH

Theorem 23 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Hermitian man-ifold119872 onto an almost paracontact pseudometric manifold119872and let the fibres of 119891 be pseudo-Riemannian submanifolds of119872 Then the submersion is an affine map on V if and only ifthe fibres of 119891 are totally geodesic

Proof We have for any 119880119881 isinV

(nabla119891lowast) (119880 119881) = minus119891

lowast(ℎ (nabla119880119881)) (89)

which in view of (27) gives

(nabla119891lowast) (119880 119881) = minus119891

lowast(T119880119881) (90)

Let the fibres of 119891 be totally geodesic Then T = 0Consequently from the above equation we have nabla119891

lowast= 0

Thus the map 119891 is affine onVConversely let the submersion 119891 be an affine map onV

Then nabla119891lowast= 0 which impliesT = 0

Hence the fibres of 119891 are totally geodesic

Theorem 24 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Hermitian man-ifold119872 onto an almost paracontact pseudometric manifold119872and let the fibres of 119891 be pseudo-Riemannian submanifoldsof 119872 Then the submersion is an affine map if and only ifℎ(nabla119864ℎ119865)+A

ℎ119864V119865+TV119864V119865 is 119891-related to nabla

119883119884 for any 119864 119865 isin

Γ(119879119872)

Proof For any 119864 119865 isin Γ(119879119872) with 119891lowastℎ119864 = 119883 ∘ 119891 and 119891

lowastV119865 =

119884 ∘ 119891 we have

(nabla119891lowast) (119864 119865) = (nabla

119891lowastℎ119864(119891lowastℎ119865)) ∘ 119891 minus 119891

lowast(ℎ (nabla119864119865))

= (nabla119883119884) ∘ 119891 minus 119891

lowast(ℎ (nablaℎ119864ℎ119865 + nabla

ℎ119864V119865

+nablaV119864ℎ119865 + nablaV119864ℎ119865))

(91)

By using (27) and (31) in the above equation we have

(nabla119891lowast) (119864 119865) = (nabla

119883119884) ∘ 119891 minus 119891

lowast(ℎ (nabla119864ℎ119865) +A

ℎ119864V119865

+TV119864V119865) (92)

Let the submersion map be affine Then for any 119864 119865 isin

Γ(119879119872) (nabla119891lowast)(119864 119865) = 0 Equation (92) implies (nabla

119883119884) ∘ 119891 =

119891lowast(ℎ(nabla119864ℎ119865) +A

ℎ119864V119865 +TV119864ℎ119865)

Conversely let ℎ(nabla119864119865) + A

ℎ119864V119865 + TV119864ℎ119865 be 119891-related

to nabla119883119884 for any 119864 119865 isin Γ(119879119872) Then from (92) we have

(nabla119891lowast)(119864 119865) = 0Hence the submersion map 119891 is affine

4 Curvature Properties

In this section the paraholomorphic bisectional curvaturesand paraholomorphic sectional curvatures of total mani-fold base manifold and fibres of paracomplex paracontactpseudo-Riemannian submersion and their curvature proper-ties are studied

Let 119891 119872 rarr 119872 be a paracomplex paracontact pseudo-Riemannian submersion from an almost para-Hermitianmanifold (119872 119869 119892) onto an almost paracontact pseudometricmanifold (119872 120601 120585 120578 119892)

Suppose that the vector fields 119864 119865 span the 2-dimensional plane at point 119901 of 119872 and let R be theRiemannian curvature tensor of 119872 The paraholomorphicbisectional curvature 119861(119864 119865) of 119872 for any pair of nonzeronon-lightlike vector fields 119864 119865 on 119872 is defined by theformula

119861 (119864 119865) =

R (119864 119869119864 119865 119869119865)

119892 (119864 119864) 119892 (119865 119865) (93)

For a nonzero non-lightlike vector field 119864 the vector field119869119864 is also non-lightlike and 119864 119869119864 span the 2-dimensionalplane Then the paraholomorphic sectional curvature 119867(119864)is defined as

119867(119864) = 119861 (119864 119864) =

R (119864 119869119864 119864 119869119864)

119892 (119864 119864) 119892 (119864 119864) (94)

The curvature properties of Riemannian submersion andsemi-Riemannian submersion have been extensively studiedin the work of OrsquoNeill [1] and Gray [3]

Let 119861ℎ and 119861V be the paraholomorphic bisectional cur-vatures of horizontal and vertical spaces respectively Let119867ℎ and 119867V be the paraholomorphic sectional curvatures of

horizontal and vertical spaces respectively Let 119861 and 119867 bethe paraholomorphic bisectional and sectional curvatures ofthe base manifold respectively

Proposition 25 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion from an almostpara-Hermitian manifold onto an almost paracontact pseudo-metric manifold and let the fibres of 119891 be pseudo-Riemanniansubmanifolds of 119872 Let 119880 119881 be non-lightlike unit verticalvector fields and let119883 119884 be non-lightlike unit horizontal vectorfields on119872 Then one has

119861 (119880119881) = 119861V(119880 119881) + 119892 (T

119880(119869119881) T

119869119880119881)

minus 119892 (T119869119880(119869119881) T

119880119881)

(95)

Geometry 11

119861 (119883119880) = 119892 ((nabla119880A)119883

119869119883 119869119880) minus 119892 (A119883119869119880A

119869119883119880)

+ 119892 (A119883119880A119869119883119869119880) minus 119892 ((nabla

119869119880A)119883

119869119883119880)

+ 119892 (T119869119880119883T119880(119869119883))

minus 119892 (T119880119883T119869119880(119869119883))

(96)

119861 (119883 119884) = 119861ℎ

(119883 119884) minus 2119892 (A119883(119869119883) A

119884(119869 119884))

+ 119892 (A119869119883119884A119883(119869 119884))

minus 119892 (A119883119884A119869119883(119869 119884))

(97)

Proof Using Definitions (93) and (94) of paraholomorphicsectional curvature and fundamental equations of submer-sion obtained by OrsquoNeill [1] we have (95) (96) and (97)

Corollary 26 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion from an almostpara-Hermitian manifold onto an almost paracontact pseudo-metric manifold If the fibres of 119891 are totally geodesic pseudo-Riemannian submanifolds of119872 then for any non-lightlike unitvertical vector fields 119880 and 119881 one has

119861 (119880119881) = 119861V(119880 119881) (98)

Corollary 27 Let 119891 119872 rarr 119872 be a paracomplex para-contact pseudo-Riemannian submersion from an almost para-Hermitian manifold onto an almost paracontact pseudometricmanifold and let the fibres of 119891 be totally geodesic pseudo-Riemannian submanifolds of119872 If the horizontal distributionis integrable then for any non-lightlike unit horizontal vectorfields119883 and 119884 one has

119861 (119883 119884) = 119861ℎ

(119883 119884) (99)

Proposition 28 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion from an almostpara-Hermitian manifold onto an almost paracontact pseudo-metric manifold and let the fibres of 119891 be pseudo-Riemanniansubmanifolds of119872 Let 119880 and 119883 be non-lightlike unit verticalvector field and non-lightlike unit horizontal vector fieldrespectively Then one has

119867(119880) = 119867V(119880) +

10038171003817100381710038171003817T119880(119869119880)

10038171003817100381710038171003817

2

minus 119892 (T119869119880(119869119880) T

119880119880)

(100)

119867(119883) = 119867 (119883) ∘ 119891 minus 310038171003817100381710038171003817A119883(119869119883)

10038171003817100381710038171003817

2

(101)

Proof The proof is straightforward If we take 119880 = 119881 in (95)and119883 = 119884 in (97) we have (98) and (99)

Corollary 29 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion from an almostpara-Hermitian manifold onto an almost paracontact pseudo-metric manifold If the fibres of 119891 are totally geodesic pseudo-Riemannian submanifolds of 119872 then the total manifold andfibres of119891have the same paraholomorphic sectional curvatures

Proof Since the fibres are totally geodesic T = 0 conse-quently we have

119867(119880) = 119867V(119880) (102)

Corollary 30 Let 119891 119872 rarr 119872 be a paracomplex para-contact pseudo-Riemannian submersion from an almost para-Hermitian manifold onto an almost paracontact pseudometricmanifold and let the fibres of 119891 be totally geodesic pseudo-Riemannian submanifolds of119872 If the horizontal distributionis integrable then the base manifold and horizontal distribu-tion have the same paraholomorphic sectional curvatures

Proof Since the horizontal distribution is integrableA = 0consequently we have

119867(119883) = 119867 (119883) ∘ 119891 (103)

Theorem 31 Let 119891 119872119898

rarr 119872119899 be a paracomplex

paracontact pseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometricmanifold 119872 and let the fibres of 119891 be pseudo-Riemanniansubmanifolds of 119872 If 119880 119881 are the non-lightlike unit verticalvector fields and 119883 119884 are the non-lightlike unit horizontalvector fields then one has

119861 (119880119881) = 119861V(119880 119881) (104)

119861 (119883119880) = minus210038171003817100381710038171003817T11988011988310038171003817100381710038171003817

2

(105)

119861 (119883 119884) = 119861 (119883 119884) ∘ 119891 (106)

Proof Using results of Lemma 19 in (95) we have

119861 (119880119881) = 119861V(119880 119881) minus 119892 (119869 (T

119880119881) 119869 (T

119880119881))

minus 119892 (1198692

(T119880119881) T

119880119881)

= 119861 (119880119881) + 119892 (T119880119881T119880119881) minus 119892 (T

119880119881T119880119881)

= 119861V(119880 119881)

(107)

Applying results of Lemma 19 in (96) we have

119861 (119883119880) = 119892 ((nabla119880A)119883

(119869119883) 119869119880) minus 119892 ((nabla119869119880A)119883

(119869119883) 119880)

+ 21003817100381710038171003817A119883119880

1003817100381710038171003817

2

minus 210038171003817100381710038171003817T11988011988310038171003817100381710038171003817

2

(108)

Since byTheorem 21 the horizontal distribution is integrablewe haveA = 0 which implies

119861 (119883119880) = minus210038171003817100381710038171003817T11988011988310038171003817100381710038171003817

2

(109)

In view ofA = 0 (104) follows from (97)

12 Geometry

Theorem 32 Let 119891 119872119898

rarr 119872119899 be a paracomplex

paracontact pseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometricmanifold 119872 and let the fibres of 119891 be pseudo-Riemanniansubmanifolds of119872 If 119880 119883 are non-lightlike unit vertical andnon-lightlike unit horizontal vector fields respectively then onehas

119867(119880) = 119867V(119880) minus 2

1003817100381710038171003817T1198801198801003817100381710038171003817

2

(110)

119867(119883) = 119867 (119883) ∘ 119891 (111)

Proof Since 119891 is the paracomplex paracontact pseudo-Riemannian submersion from a para-Kahler manifold 119872

onto an almost paracontact pseudometric manifold 119872 by(16) and equations of Lemma 19 we have

119892 (T119869119880(119869119880) T

119880119880) = 119892 (119869

2

(T119880119880) T

119880119880) =

1003817100381710038171003817T1198801198801003817100381710038171003817

2

119892 (T119880(119869119880) T

119880(119869119880)) = minus119892 (T

119880119880T119880119880) = minus

1003817100381710038171003817T1198801198801003817100381710038171003817

2

(112)

and by using the above results in (100) we have

119867(119880) = 119867V(119880) minus

1003817100381710038171003817T1198801198801003817100381710038171003817

2

minus1003817100381710038171003817T119880119880

1003817100381710038171003817

2

= 119867V(119880) minus 2

1003817100381710038171003817T1198801198801003817100381710038171003817

2

(113)

Again since horizontal distribution is integrable we haveA = 0 and putting it in (101) we obtain (111)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Uma Shankar Verma is thankful to University Grant Com-mission New Delhi India for financial support

References

[1] B OrsquoNeill ldquoThe fundamental equations of a submersionrdquo TheMichigan Mathematical Journal vol 13 pp 459ndash469 1966

[2] B OrsquoNeill Semi-Riemannian Geometry with Applications toRelativity vol 103 of Pure and Applied Mathematics AcademicPress New York NY USA 1983

[3] A Gray ldquoPseudo-Riemannian almost product manifolds andsubmersionsrdquo vol 16 pp 715ndash737 1967

[4] J P Bourguignon and H B Lawson A Mathematicians Visitto Kaluza-Klein Theory Rendiconti del Seminario Matematico1989

[5] S Ianus and M Visinescu ldquoSpace-time compactification andRiemannian submersionsrdquo in The Mathematical Heritage GRassias and C F Gauss Eds pp 358ndash371 World ScientificRiver Edge NJ USA 1991

[6] J P Bourguignon and H B Lawson Jr ldquoStability and isolationphenomena for Yang-Mills fieldsrdquo Communications in Mathe-matical Physics vol 79 no 2 pp 189ndash230 1981

[7] B Watson ldquoG 1198661015840

-Riemannian submersions and nonlineargauge field equations of general relativityrdquo in Global AnalysismdashAnalysis on Manifolds T Rassias and M Morse Eds vol 57 ofTeubner-Texte zur Mathematik pp 324ndash349 Teubner LeipzigGermany 1983

[8] C Altafini ldquoRedundant robotic chains on Riemannian submer-sionsrdquo IEEE Transactions on Robotics and Automation vol 20no 2 pp 335ndash340 2004

[9] M T Mustafa ldquoApplications of harmonic morphisms to grav-ityrdquo Journal of Mathematical Physics vol 41 no 10 pp 6918ndash6929 2000

[10] B Watson ldquoAlmost Hermitian submersionsrdquo Journal of Differ-ential Geometry vol 11 no 1 pp 147ndash165 1976

[11] D Chinea ldquoAlmost contact metric submersionsrdquo Rendiconti delCircolo Matematico di Palermo II vol 34 no 1 pp 319ndash3301984

[12] D Chinea ldquoTransference of structures on almost complexcontact metric submersionsrdquo Houston Journal of Mathematicsvol 14 no 1 pp 9ndash22 1988

[13] Y Gunduzalp and B Sahin ldquoPara-contact para-complexpseudo-Riemannian submersionsrdquo Bulletin of the MalaysianMathematical Sciences Society

[14] I Sato ldquoOn a structure similar to the almost contact structurerdquoTensor vol 30 no 3 pp 219ndash224 1976

[15] P K Rashevskij ldquoThe scalar field in a stratified spacerdquoTrudy Seminara po Vektornomu i Tenzornomu Analizu s ikhPrilozheniyami k Geometrii Mekhanike i Fizike vol 6 pp 225ndash248 1948

[16] P Libermann ldquoSur les structures presque paracomplexesrdquoComptes Rendus de lrsquoAcademie des Sciences I vol 234 pp 2517ndash2519 1952

[17] E M Patterson ldquoRiemann extensions which have Kahlermetricsrdquo Proceedings of the Royal Society of Edinburgh AMathematics vol 64 pp 113ndash126 1954

[18] S Sasaki ldquoOn differentiable manifolds with certain structureswhich are closely related to almost contact structure Irdquo TheTohoku Mathematical Journal vol 12 pp 459ndash476 1960

[19] S Zamkovoy ldquoCanonical connections on paracontact mani-foldsrdquo Annals of Global Analysis and Geometry vol 36 no 1pp 37ndash60 2009

[20] D E Blair Riemannian Geometry of Contact and SymplecticManifolds vol 23 of Progress in Mathematics BirkhauserBoston Mass USA 2002

[21] V Cruceanu P Fortuny and P M Gadea ldquoA survey onparacomplex geometryrdquoThe Rocky Mountain Journal of Mathe-matics vol 26 no 1 pp 83ndash115 1996

[22] P M Gadea and J M Masque ldquoClassification of almost para-Hermitian manifoldsrdquo Rendiconti di Matematica e delle sueApplicazioni VII vol 7 no 11 pp 377ndash396 1991

[23] M Falcitelli S Ianus and A M Pastore Riemannian Submer-sions and Related Topics World Scientific River Edge NJ USA2004

[24] E G Rio and D N Kupeli Semi-Riemannian Maps andTheir Applications Kluwer Academic Publisher DordrechtTheNetherlands 1999

[25] B Sahin ldquoSemi-invariant submersions from almost HermitianmanifoldsrdquoCanadianMathematical Bulletin vol 54 no 3 2011

Submit your manuscripts athttpwwwhindawicom

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 4: Research Article Paracomplex Paracontact Pseudo-Riemannian …downloads.hindawi.com/archive/2014/616487.pdf · 2019-07-31 · Research Article Paracomplex Paracontact Pseudo-Riemannian

4 Geometry

ℎ (nabla119880119883) = ℎ (nabla

119883119880) = A

119883119880 (35)

A119883119884 =

1

2V [119883 119884] (36)

A119883119884 = minusA

119884119883 (37)

T119880119881 = T

119881119880 (38)

for all 119864 119865 isin Γ(119879119872)MoreoverT

119880119881 coincideswith second fundamental form

of the submersion of the fibre submanifoldsThe distributionH is completely integrable In view of (37) and (38) A isalternating on the horizontal distribution andT is symmetricon the vertical distribution

3 Paracomplex Paracontact Pseudo-Riemannian Submersions

In this section we introduce the notion of pseudo-Riemannian submersion from almost paracomplex mani-folds onto almost paracontact pseudometric manifolds illus-trate examples and study the transference of structures ontotal manifolds and base manifolds

Definition 2 Let (1198722119898 119869 119892) be an almost para-Hermitianmanifold and let (1198722119899+1 120601 120585 120578 119892) be an almost paracontactpseudometric manifold

A pseudo-Riemannian submersion 119891 119872 rarr 119872 is calledparacomplex paracontact pseudo-Riemannian submersion ifthere exists a 1-form 120578 on119872 such that

119891lowast∘ 119869 = 120601 ∘ 119891

lowast+ 120578 otimes 120585 (39)

Since for each 119901 isin 119872119891lowast119901

is a linear isometry betweenhorizontal spacesH

119901and tangent spaces119879

119891(119901)119872 there exists

an induced almost paracontact structure (120601ℎ

120578ℎ

120585ℎ

119892) on(2119899 + 1)-dimensional horizontal distribution H such that120601ℎ

|

Dℎbehave just like the fundamental collineation of almost

paracomplex structure 119869 on ker 120578ℎ = Dℎ

and 120601ℎ

Dℎ

rarr Dℎ

is an endomorphism such that 120601ℎ

= 119869|ker 120578ℎ

and the rank of

120601ℎ

= 2119899 where dim(Dℎ

) = 2119899It follows that for any 119883ℎ isin D

ℎ

120578ℎ(119883ℎ) = 0 whichimplies that 1198692

|

Dℎ(119883ℎ

) = (120601ℎ

)2

(119883ℎ

) = 119883ℎ for any 119883ℎ isin D

ℎ

andH = Dℎ

oplus 120585ℎ

[18]

Definition 3 (see [25]) A pseudo-Riemannian submersion119891 119872 rarr 119872 is called semi-119869-invariant submersion if thereis a distributionD

1sube ker119891

lowastsuch that

ker119891lowast= D1oplusD2 (40)

119869 (D1) = D

1 119869 (D

2) sube (ker119891

lowast)perp

(41)

whereD2is orthogonal complementary toD

1in ker119891

lowast

Proposition 4 Let119891 1198722119898 rarr 1198722119899+1 be a paracomplex par-

acontact pseudo-Riemannian submersion and let the fibres of119891 be pseudo-Riemannian submanifolds of119872 Then the fibres119891minus1

(119902) 119902 isin 119872 are semi-119869-invariant submanifolds of 119872 ofdimension (2119898 minus 2119899 minus 1)

Proof Let 119880 isinV Then

119891lowast(119869119880) = 120601 (119891

lowast(119880)) + 120578 (119880) 120585

997904rArr 119891lowast119869 (119880) minus 120578 (119880) 120585

ℎ

= 0

(42)

where 119891lowast120585ℎ

= 120585Thus we have

119869 (119880) minus 120578 (119880) 120585ℎ

= 120601 (119880) for some 120601 (119880) isinV (43)

By (19) we get 119892(120585ℎ

119869(120585ℎ

)) = 0 = 119892(120585 119891lowast(119869(120585ℎ

))) = 0As 119892 is nondegenerate on119872 we have

119891lowast(119869 (120585ℎ

)) = 0 that is 119869 (120585ℎ

) isinV (44)

Taking 119880 = 119869120585ℎ

in (43) we obtain

120585ℎ

minus 120578 (119869120585ℎ

) 120585ℎ

= 120601(119869120585ℎ

) (45)

Since fibre 119891minus1(119902) is an odd dimensional submanifold thereexists an associated 1-form 120578

V which is restriction of 120578 on fibresubmanifold 119891minus1(119902) 119902 isin 119872 and a characteristic vector field120585V= 119869120585ℎ

such that 120601(120585V) = 0 So we have 120578V(120585

V) = 1

Let us put ker 120578V = D1andD

2= 120585

V

Then ker119891lowast= D1oplus D2and 119869(D

1) = D

1 119869(D

2) =

119869120585V = 120585

ℎ

sube (ker119891lowast)perp

Hence the fibres 119891minus1(119902) are semi-119869-invariant submani-folds of119872

Corollary 5 Let 119891 1198722119898

rarr 1198722119899+1 be a paracomplex

paracontact pseudo-Riemannian submersion and let the fibresof 119891 be pseudo-Riemannian submanifolds of 119872 Then thefibres 119891

minus1

(119902) are almost paracontact pseudometric mani-folds with almost paracontact pseudo-Riemannian structures(120601

V 120585

V 120578

V 119892

V) 119902 isin 119872 where 120585

V= 119869(120585

ℎ

) 120578V = 120578|V and

119892V= 119892

Proof Since 119891minus1(119902) are semi-119869-invariant submanifolds of119872of odd dimension 2119903 + 1 = 2119898 minus 2119899 minus 1 (39) implies

119869 (119880) = 120601V119880 + 120578

V(119880) 120585ℎ

(46)

for any 119880 isinV

Geometry 5

On operating 119869 on both sides of the above equation weget

119880 = 120601V(120601

V(119880)) + 120578

V(120601

V(119880)) 120585

ℎ

+ 120578V(119880) 120585

V (47)

where 119869(120585ℎ

) = 120585V

Equating horizontal and vertical components we have

119880 = 120601V(120601

V(119880)) + 120578

V(119880) 120585

V 120578

V∘ 120601

V(119880) = 0

997904rArr (120601V)2

(119880) = 119880 minus 120578V(119880) 120585

V 120578

V∘ 120601

V= 0

120601V(120585

V) = 0 120578

V(120585

V) = 1

(48)

Hence (120601V 120585

V 120578

V 119892

V) is almost paracontact pseudometric

structure on the fibre 119891minus1(119902) 119902 isin 119872

Proposition 6 Let 119891 1198722119898

rarr 1198722119899+1 be a paracomplex

paracontact pseudo-Riemannian submersion and let the fibresof 119891 be pseudo-Riemannian submanifolds of 119872 Let 119883 119884 bebasic vector fields 119891-related to 119883 119884 respectively Let 120578 and 120578be 1-forms on the total manifold119872 and the base manifold119872respectively Then one has the following

(i) The characteristic vector field 119869120585ℎ

is a vertical vectorfield

(ii) 119891lowastlowast120578 = 120578ℎ where 119891lowast

lowast120578 is pullback of 120578 through 119891

lowast

(iii) 120578ℎ(119880) = 0 for any vertical vector field 119880(iv) 120578V(119883) = 0 for any horizontal vector field119883

Remark 7 Results (ii) and (iv) are analogue version of results(i) and (iii) of Proposition 4 of [13]

Proof (i) By Corollary 5 (120601V 120585

V 120578

V 119892

V) is almost paracontact

pseudometric structure on 119891minus1(119902) We have

0 = 119892 (120585ℎ

119869120585ℎ

) = 119892(119891lowast(120585ℎ

) 119891lowast(119869120585ℎ

))

= 119892(120585 119891lowast(119869120585ℎ

))

(49)

Now

119891lowast(119869120585ℎ

) = 120601 ∘ 119891lowast120585ℎ

+ 120578 (120585ℎ

) 120585 = 120578 (120585ℎ

) 120585 (50)

so we have

0 = 119892(120585 120578 (120585ℎ

) 120585) = 120578 (120585ℎ

)119892 (120585 120585) = 120578 (120585ℎ

) (51)

Thus 119891lowast(119869120585ℎ

) = 0Hence 119869120585

ℎ

is a vertical vector field(ii) Since 119891 119872

2119898

rarr 1198722119899+1 is smooth submersion

120578ℎ

= 120578|H

is restriction of 120578 on the horizontal distribution

H and 119891lowast119901

H119901rarr 119879119891(119901)

119872 is a linear isometry for any119883119901isinH119901 we get

120578ℎ

119901(119883119901) = 120576119892

119901(120585ℎ

119901 119883119901) = 119892

119891(119901)(119891lowast119901120585ℎ

119901 119891lowast119901119883119901)

= 119892119891(119901)

(120585119891(119901)

119883119891(119901)

) = 120578119891(119901)

(119883119891(119901)

) = 119891lowast

lowast120578119901(119883119901)

(52)

Hence pullback 119891lowastlowast120578 = 120578ℎ

Results (iii) and (iv) immediately follow from the previ-ous results

Example 8 Let (R42 119869 119892) be a paracomplex pseudometric

manifold and let (R31 120601 120585 120578 119892) be an almost paracontact

pseudometric manifoldDefine a submersion 119891 R4

2 (1199091 1199092 1199101 1199102)119905

rarr

R31 (119906 V 119908)119905 by

119891 ((1199091 1199092 1199101 1199102)119905

) 997891997888rarr (1199091+ 1199092+ 31199101+ 21199102

31199091+ 21199092+ 1199101+ 1199102

51199091+ 31199092+ 51199101+ 31199102)119905

(53)

Then the kernel of 119891lowastis

V = ker119891lowast= Span119881

1=

120597

1205971199091

minus 2120597

1205971199092

minus120597

1205971199101

+ 2120597

1205971199102

(54)

which is the vertical distribution admitting one lightlikevector field that is fibre is degenerate submanifold of R4

2

The horizontal distribution is

H = (ker119891lowast)perp

= Span1198831=

120597

1205971199091

minus120597

1205971199101

1198832=

120597

1205971199092

+ 2120597

1205971199101

1198833= 2

120597

1205971199101

+120597

1205971199102

(55)

For any real 119896 the horizontal characteristic vector field 120585ℎ

isgiven by

120585ℎ

= 119896120597

1205971199091

minus (2119896 minus1

3)

120597

1205971199092

minus (119896 minus 1)120597

1205971199101

+ (2119896 minus5

3)

120597

1205971199102

(56)

which is 119891-related to the characteristic vector field 120585 = 120597120597119908Moreover there exists one form 120578 = 5119889119909

1+3119889119909

2+51198891199101+

31198891199102on (R4

2 119869 119892) such that the submersion satisfies (39)

Example 9 Let (R63 119869 119892) be an almost paracomplex pseudo-

Riemannian manifold and let (R31 120601 120585 120578 119892) be an almost

6 Geometry

paracontact pseudo-Riemannian manifold Consider a sub-mersion 119891 R6

3 (1199091 1199092 1199093 1199101 1199102 1199103)119905

rarr R31 (119906 V 119908)119905

defined by

119891 ((1199091 1199092 1199093 1199101 1199102 1199103)119905

)

997891997888rarr (1199091+ 1199092

radic2

1199101+ 1199102

radic2

1199102+ 1199103

radic2

)

119905

(57)

Then there exists one form 120578 = (1198891199092+ 1198891199093)radic2 on (R6

3 119869 119892)

such that (39) is satisfied The kernel of 119891lowastis

V = ker119891lowast

= Span1198811=

120597

1205971199091

minus120597

1205971199092

1198812=

120597

1205971199101

minus120597

1205971199102

+120597

1205971199103

1198813=

120597

1205971199093

(58)

which is vertical distribution admitting non-lightlike vectorfields that is the fibre is nondegenerate submanifold of(R63 119869 119892)The horizontal distribution is

H = Span1198831=

120597

1205971199091

+120597

1205971199092

1198832= minus

120597

1205971199101

+120597

1205971199103

1198833=

120597

1205971199101

+120597

1205971199102

(59)

Example 10 Let (R42 119869 119892) be a paracomplex pseudometric

manifold and let (R31 120601 120585 120578 119892) be an almost paracontact

pseudometric manifoldConsider a submersion 119891 R4

2 (1199091 1199092 1199101 1199102)119905

rarr R31

(119906 V 119908)119905 defined by

119891 ((1199091 1199092 1199101 1199102)119905

) 997891997888rarr (1199091 1199101 1199102)119905

(60)

Then the kernel of 119891lowastis

V = ker119891lowast= Span119881

1=

120597

1205971199092

(61)

which is the vertical distribution and the restriction of 119892 tothe fibres of 119891 is nondegenerate

The horizontal distribution is

H = (ker119891lowast)perp

= Span119883 =120597

1205971199091

119884 =120597

1205971199101

120585ℎ

=120597

1205971199102

(62)

The characteristic vector field 120585 = 120597120597119908 on R31has unique

horizontal lift 120585ℎ

which is the characteristic vector field onhorizontal distributionH of R4

2

We also have

119892 (119883119883) = 119892 (119891lowast119883119891lowast119883) = minus1

119892 (119884 119884) = 119892 (119891lowast119884 119891lowast119884) = 1

119892 (120585ℎ

120585ℎ

) = 119892(119891lowast120585ℎ

119891lowast120585ℎ

) = 119892 (120585 120585) = 1

(63)

Thus the smooth map 119891 is a pseudo-Riemannian submer-sion

Moreover we obtain that there exists a 1-form 120578 = 1198891199092on

R42such that 120578(119869120585

ℎ

) = 1 120578(120585ℎ

) = 0 and the map 119891 satisfies

119891lowast119869119883 = 120601119891

lowast119883 + 120578 (119883) 120585

119891lowast119869119884 = 120601119891

lowast119884 + 120578 (119884) 120585

119891lowast119869120585ℎ

= 120601119891lowast120585ℎ

+ 120578 (120585ℎ

) 120585

(64)

Hence the map 119891 is a paracomplex paracontact pseudo-Ri-emannian submersion from R4

2on to R3

1

Proposition 11 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion and let the fibresof 119891 be pseudo-Riemannian submanifolds of 119872 Let 119883 119884 bebasic vector fields 119891-related to119883 119884 respectively Then 119869(119883) minus120576119892(119883 120585

ℎ

)120585ℎ

is 119891-related to 120601119883

Proof Since119883 is 119891-related to vector field119883 on119872 we have

120578 (119883) = 120578V+ 120578ℎ

(119883) = 0 + 120578ℎ

(119883) = 120576119892 (119883 120585ℎ

)

997904rArr 119891lowast(119869119883) = 120601119883 + 120578

ℎ

(119883) 120585

997904rArr 119891lowast119869119883 minus 120576119892 (119883 120585

ℎ

) 120585ℎ

= 120601119883

(65)

Hence 119869(119883) minus 120576119892(119883 120585ℎ

)120585ℎ

is 119891-related to 120601119883

Proposition 12 Let 119891 119872 rarr 119872 be a paracomplex para-contact pseudo-Riemannian submersion and let the fibres of 119891be pseudo-Riemannian submanifolds of 119872 Let V and H bethe vertical and horizontal distributions respectively If 120585

ℎ

isthe basic characteristic vector field of horizontal distribution119891-related to the characteristic vector field 120585 of base manifoldthen

(i) 119869V sub DVoplus 119869120585ℎ

oplus 120585ℎ

(ii) 119869H sub Dℎ

oplus 120585ℎ

oplus 119869120585ℎ

Proof (i) Let 119880 isin V Then 119880 = 119886119880|DV + 119887119869120585

ℎ

for 119886 119887 isin

119862infin

(119872) as 119869120585ℎ

= 120585Vis characteristic vector field on odd

Geometry 7

dimensional fibre submanifold 119891minus1(119902) of119872 119902 isin 119872 We get

119869119880 = 119886119869119880|DV + 119887119869

2

120585ℎ

= 119886119869119880|DV + 119887120585

ℎ

isinV oplus 120585ℎ

997904rArr 119869V subV oplus 120585ℎ

(66)

Again let 119881 isinV oplus 120585ℎ

Then 119881 = 119886119881|DV + 119887119869120585

ℎ

+ 119888120585ℎ

where

120578V(119881|DV ) = 0 D

V= ker 120578V 119886119881

|DV + 119887119869120585

ℎ

isin V and 119886 119887 119888 isin119862infin

(119872) We have

119869119881 = 119886119869119881|DV + 119887120585

ℎ

+ 119888119869120585ℎ

= (119886119869119881|DV + 119888119869120585

ℎ

)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

isinV

+ 119887120585ℎ

⏟⏟⏟⏟⏟⏟⏟

isin120585

ℎ

isinV oplus 120585ℎ

(67)

Now by (39) we get

119891lowast119869119881 = 120601 (119891

lowast119881) + 120578 (119891

lowast(119881)) 120585

= 119888 120601 (119891lowast120585ℎ

) + 120578 (120585) 120585

= 119888120585 isin 120585 sube 119869V

(68)

We get 119869119881 notinVHence 119869V subV oplus 120585

ℎ

that is 119869V sub DVoplus 119869120585ℎ

oplus 120585ℎ

(ii) Let 119883 = 119886119883

|

Dℎ+ 119887120585ℎ

isin H where H = Dℎ

oplus

120585ℎ

ker 120578ℎ = Dℎ

and 119886 119887 isin 119862infin(119872) Then

119869119883 = 119886119869119883|

Dℎ+ 119887119869120585ℎ

isinH oplus 119869120585ℎ

(69)

which implies that 119869H subH oplus 119869120585ℎ

Again let119884 isinHoplus119869120585

ℎ

Then119884 = 119886119884|

Dℎ+119887120585ℎ

+119888119869120585ℎ

notinHfor 119886 119887 119888 isin 119862infin(119872) We have

119869 119884 = 119886119869119884|

Dℎ+ 119887119869120585ℎ

+ 1198881198692

120585ℎ

= 119886119869119884|

Dℎ+ 119887119869120585ℎ

+ 119888120585ℎ

= 119885 + 119887119869120585ℎ

isinH oplus 119869120585ℎ

for some 119885 = 119886119869119884|

Dℎ+ 119888120585ℎ

isinH

(70)

We obtain 119869 119884 notinHHence 119869H sub H oplus 119869(120585

ℎ

) that is 119869H sub Dℎ

oplus 120585ℎ

oplus

119869120585ℎ

Example 13 Let (R63 119869 119892) be an almost paracomplex pseudo-

Riemannian manifold and let (R31 120601 120585 120578 119892) be an almost

paracontact pseudo-Riemannian manifold Consider a sub-mersion 119891 R6

3 (1199091 1199092 1199093 1199101 1199102 1199103)119905

rarr R31 (119906 V 119908)119905

defined by

119891 ((1199091 1199092 1199093 1199101 1199102 1199103)119905

) 997891997888rarr (1199091+ 1199092

radic2

1199101+ 1199102

radic2

1199103)

119905

(71)

Then the kernel of 119891lowastis

V = ker119891lowast

= Span1198811=

120597

1205971199091

minus120597

1205971199092

1198812=

120597

1205971199101

minus120597

1205971199102

120585V=

120597

1205971199093

(72)

which is the vertical distribution and the restriction of 119892 tothe fibres of 119891 is nondegenerate

The horizontal distribution is

H = (ker119891lowast)perp

= Span1198831=

120597

1205971199091

+120597

1205971199092

1198832=

120597

1205971199101

+120597

1205971199102

120585ℎ

=120597

1205971199103

(73)

The characteristic vector field 120585 = 120597120597119908 on R31has unique

horizontal lift 120585ℎ

which is the characteristic vector field onthe horizontal distributionH of R6

3

We also have

119892 (1198831 1198831) = 119892 (119891

lowast1198831 119891lowast1198831) = minus2

119892 (1198832 1198832) = 119892 (119891

lowast1198832 119891lowast1198832) = 2

119892 (120585ℎ

120585ℎ

) = 119892(119891lowast120585ℎ

119891lowast120585ℎ

) = 119892 (120585 120585) = 1

(74)

Thus the smooth map 119891 is a pseudo-Riemannian submer-sion

Also we obtain that there exists a 1-form 120578 = 1198891199093on R63

such that 120578(119869120585ℎ

) = 1 120578(120585ℎ

) = 0 and the map 119891 satisfies

119891lowast1198691198831= 120601119891lowast1198831+ 120578 (119883

1) 120585

119891lowast1198691198832= 120601119891lowast1198832+ 120578 (119883

2) 120585

119891lowast119869120585ℎ

= 120601119891lowast120585ℎ

+ 120578 (120585ℎ

) 120585

(75)

Hence the map 119891 is a paracomplex paracontact pseudo-Riemannian submersion from R6

3onto R3

1

Moreover we observe that for this submersion119891 we have

119869V subV oplus 120585ℎ

119869H subH oplus 119869120585ℎ

(76)

which verifies Proposition 12

8 Geometry

Proposition 14 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion and let the fibresof 119891 be pseudo-Riemannian submanifolds of 119872 Let 119883 119884 bebasic vector fields 119891-related to 119883 119884 respectively Let 119865 and Φbe the second fundamental forms and let nabla and nabla be the Levi-Civita connection on the total manifold119872 and base manifold119872 respectively Then one has

(i) 119891lowast((nabla119883119869)119884) = (nabla

119883120601)119884 + 120576119892(119884 nabla

119883120585)120585 + 120578(119884)nabla

119883120585

(ii) 119865 = 119891lowastlowastΦ + 120576120578 otimes 120578

(iii) 119891lowast((nabla119883119865)(119884 119885)) = (nabla

119883Φ)(119884 119885) + 120578(119884)119892(119885 nabla

119883120585) +

120578(119885)119892(119884 nabla119883120585)

Proof (i) In view of Definition 2 and Proposition 11 we have

119891lowast((nabla119883119869) 119884) = 119891

lowast(nabla119883(119869 119884) minus 119869 (nabla

119883119884))

= nabla119883(119891lowast(119869 119884)) minus 119891

lowast(119869 (nabla119883119884))

= nabla119883(120601119884) + nabla

119883(120578 (119884) 120585) minus 120601 (nabla

119883119884)

minus 120578 (nabla119883119884) 120585

= (nabla119883120601)119884 + nabla

119883(120576119892 (119884 120585) 120585) minus 120578 (nabla

119883119884) 120585

= (nabla119883120601)119884 + 120576119892 (nabla

119883119884 120585) 120585 + 120576119892 (119884 nabla

119883120585) 120585

+ 120576119892 (119884 120585) nabla119883120585 minus 120576119892 (nabla

119883119884 120585) 120585

= (nabla119883120601)119884 + 120576119892 (119884 nabla

119883120585) 120585 + 120578 (119884) nabla

119883120585

(77)

(ii) Since119891lowastlowastΦ is pullback ofΦ through the linearmap119891

lowast

we get

119891lowast

lowastΦ(119883 119884) = Φ (119883 119884) ∘ 119891 = 119892 (119883 120601119884) ∘ 119891

= 119892 (119883 119869 119884) minus 120576120578 (119884) 120578 (119883)

= 119865 (119883 119884) minus 120576120578 (119883) 120578 (119884)

(78)

which implies 119865 = 119891lowastlowastΦ + 120576120578 otimes 120578

(iii) By (23) we have

119891lowast((nabla119883119865) (119884 119885)) = 119892 (119891

lowast(119884) 119891

lowast((nabla119883119869)119885)) (79)

Now using (i) in the above equation we get (iii)

Theorem 15 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion and let the fibresof 119891 be pseudo-Riemannian submanifolds of 119872 Let 119883 119884 bebasic vector fields 119891-related to 119883 119884 respectively If the totalspace is para-Hermitian manifold then the almost paracontactstructure of base space is normal

Moreover if the almost paracontact structure of base spaceis normal then the Nijenhuis tensor of total space is vertical

Proof The Nijenhuis tensors 119873119869and 119873

120601of almost para-

complex structure 119869 and almost paracontact structure 120601 arerespectively defined by (8) and (11)

Using Definition 2 and properties of Sections 21 and 22we get the following identity

119891lowast(119873119869(119883 119884)) = 119873

(1)

(119883 119884) + 2119889120578 (120601119883 119884) 120585

minus 2119889120578 (120601119884119883) 120585

+ 2120578 (119883) 119889120578 (120585 119884) 120585 minus 2120578 (119884) 119889120578 (120585 119883) 120585

minus 120578 (119884)119873(3)

(119883) + 120578 (119883)119873(3)

(119884)

(80)

Using (12) (13) (14) and (15) (80) reduces to

119891lowast(119873119869(119883 119884)) = 119873

(1)

(119883 119884) + 119873(2)

(119883 119884) 120585

+ 120578 (119883)119873(4)

(119884) 120585

minus 120578 (119884)119873(4)

(119883) 120585 minus 120578 (119884)119873(3)

(119883)

+ 120578 (119883)119873(3)

(119884)

(81)

Since 119873119869(119883 119884) = 0 it follows from (81) that

tensors 119873(1) 119873(2) 119873(3) and 119873(4) vanish togetherHence the almost paracontact structure of base space is

normalConversely let the almost paracontact structure of the

base space be normalThen (81) implies that 119891

lowast(119873119869(119883 119884)) = 0

Hence119873119869(119883 119884) is vertical

Corollary 16 Let119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion and let the fibres of 119891 bepseudo-Riemannian submanifolds of 119872 Let 119883 119884 be basicvector fields 119891-related to 119883 119884 respectively Let the total spacebe para-Hermitian manifold and119873(1) vanishes Then the basespace is paracontact pseudometric manifold if and only if 120585 iskilling

Proof Let the total space be para-Hermitian and 119873(1) van-ishes Then from (80) we have

0 = 2119889120578 (120601119883 119884) 120585 minus 2119889120578 (120601119884119883) 120585 + 2120578 (119883) 119889120578 (120585 119884) 120585

minus 2120578 (119884) 119889120578 (120585 119883) 120585 minus 120578 (119884) (L120585120601)119883 + 120578 (119883) (L

120585120601)119884

(82)

If 120585 is killing then we have L120585120601 = 0 It immediately follows

from (82) that

119889120578 (120601119883 119884) minus 120578 (120601119884119883) + 120578 (119883) 119889120578 (120585 119884)

minus 120578 (119884) 119889120578 (120585 119883) = 0

(83)

In view of (6) and (7) the above equation gives 119889120578 = ΦConversely let the base space be paracontact Then 119889120578 =

ΦUsing (6) (7) and (82) we getL

120585120601 = 0

Hence the characteristic vector field 120585 is killing

Geometry 9

Theorem 17 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion and let the fibres of 119891 bepseudo-Riemannian submanifolds of 119872 Let 119883 119884 be basicvector fields 119891-related to 119883 119884 respectively If the total spaceis para-Kahler then the base space is paracosymplectic Theconverse is true if nabla

119883119869 is vertical

Proof We have for any 119883119884 isin Γ(119879119872) (nabla119883120601)119884 = 0 which

gives 119892(119885 (nabla119883120601)119884) = 0 for any 119885 isin Γ(119879119872)

From Proposition 14 we have

119892 (119885 (nabla119883120601)119884) + 120576120578 (119884) (nabla

119883120578)119885 + 120576120578 (119885) (nabla

119883120578) 119884 ∘ 119891

= 119892 (119885 (nabla119883119869) 119884)

(84)

Let nabla 119869 = 0 that is the total space is para-KahlerThen from(84) we obtain nabla120601 = 0 and nabla120578 = 0 Hence the base space isparacosymplectic

Again let (nabla119883120601)119884 = 0 and nabla120578 = 0 Then 119892(119885 (nabla

119883119869)119884) =

0 which implies that (nabla119883119869)119884 is a vertical vector field

Theorem 18 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion and let the fibres of 119891 bepseudo-Riemannian submanifolds of 119872 Let 119883 119884 and 119885 bebasic vector fields 119891-related to 119883 119884 and 119885 respectively If120578(119883)119885(120578(119884))+120578(119884)119883(120578(119885))minus120578(119883)119884(120578(119885)) = 0 then the totalspace is almost para-Kahler if and only if the base space119872 isan almost paracosymplectic manifold

Proof We have the following equation

3119889119865 (119883 119884 119885)

= 3 (119891lowast

lowast119889Φ) (119883 119884 119885) + 2120576120578 (119885) 119889120578 (119883 119884)

minus 2120576120578 (119884) 119889120578 (119883 119885)

+ 2120576120578 (119883) 119889120578 (119884 119885) + 2120576120578 (119883)119885 (120578 (119884))

+ 2120576120578 (119884)119883 (120578 (119885))

minus 2120576120578 (119883)119884 (120578 (119885))

(85)

If 119889120578 = 0 119889Φ = 0 and 120578(119883)119885(120578(119884)) + 120578(119884)119883(120578(119885)) minus

120578(119883)119884(120578(119885)) = 0 then from (85) we have 119889119865 = 0 Hencethe total space is almost para-Kahler

Conversely let 119889119865 = 0 and 120578(119883)119885(120578(119884))+120578(119884)119883(120578(119885))minus120578(119883)119884(120578(119885)) = 0

By using the above equation in (85) we have 119889120578 = 0 and119889Φ = 0

Hence the base space is almost paracosymplectic

Nowwe investigate the properties of fundamental tensorsT andA of a pseudo-Riemannian submersion

Lemma 19 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometric manifold119872 and

let the fibres of 119891 be pseudo-Riemannian submanifolds of 119872Then for any horizontal vector fields119883 119884 and for any verticalvector fields 119880 119881 on119872 one has

(i) A119883(119869 119884) = 119869(A

119883119884)

(ii) A119869119883(119884) = 119869(A

119883119884)

(iii) T119880(119869 119881) = 119869(T

119880119881)

(iv) T119869119880119881 = 119869(T

119880119881)

Proof The proof follows using similar steps as in Lemmas 3and 4 of [13] so we omit it

Lemma 20 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometric manifold119872 andlet the fibres of 119891 be pseudo-Riemannian submanifolds of 119872Then for any vector fields 119864 119865 on119872 one has

(i) A119864(119869119865) = 119869(A

119864119865)

(ii) T119864(119869119865) = 119869(T

119864119865)

Proof The proof follows from (37) and (38)

Theorem 21 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometric manifold119872 andlet the fibres of 119891 be pseudo-Riemannian submanifolds of 119872Then the horizontal distribution is integrable

Proof For any vertical vector field 119880 we have

119892 (119869 (A119883119884) 119880) = 119892 (A

119883119869 119884119880)

= minus119892 (119869119884A119883119880)

= minus119892 (119869119884 ℎ (nabla119880119883))

= 119892 (119884 ℎ (119869 (nabla119880119883)))

= 119892 (119884 ℎ (minusnabla119880119869)119883

+nabla119880(119869119883))

= 119892 (119884 ℎ nabla119880(119869119883)) = 119892 (119884A

119869119883119880)

= minus119892 (A119869119883119884119880) = minus119892 (119869 (A

119883119884) 119880)

(86)

Thus 119892(119869(A119883119884) 119880) = 0 which is true for all119883 and 119884

SoA119883119884 = 0

Hence the horizontal distribution is integrable

Theorem 22 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometric manifold119872 andlet the fibres of 119891 be pseudo-Riemannian submanifolds of 119872Then the submersion is an affine map onH

10 Geometry

Proof The second fundamental form of 119891 is defined by

(nabla119891lowast) (119864 119865) = (nabla

119891

119864119891lowast(119865)) ∘ 119891 minus 119891

lowast(nabla119864119865) (87)

where 119864 119865 isin Γ(119879119872) and nabla119891 is pullback connection of Levi-Civita connection nabla on119872 with respect to 119891

We have for any119883119884 isinH

(nabla119891lowast) (119883 119884) = (nabla

119891

119883

119891lowast(119884)) ∘ 119891 minus 119891

lowast(nabla119883119884) (88)

By using Lemma 1 we have 119891lowast(ℎ(nabla119883119884)) = (nabla

119883119884) ∘ 119891 which

implies nabla119891lowast= 0

Hence the submersion 119891 is an affine map onH

Theorem 23 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Hermitian man-ifold119872 onto an almost paracontact pseudometric manifold119872and let the fibres of 119891 be pseudo-Riemannian submanifolds of119872 Then the submersion is an affine map on V if and only ifthe fibres of 119891 are totally geodesic

Proof We have for any 119880119881 isinV

(nabla119891lowast) (119880 119881) = minus119891

lowast(ℎ (nabla119880119881)) (89)

which in view of (27) gives

(nabla119891lowast) (119880 119881) = minus119891

lowast(T119880119881) (90)

Let the fibres of 119891 be totally geodesic Then T = 0Consequently from the above equation we have nabla119891

lowast= 0

Thus the map 119891 is affine onVConversely let the submersion 119891 be an affine map onV

Then nabla119891lowast= 0 which impliesT = 0

Hence the fibres of 119891 are totally geodesic

Theorem 24 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Hermitian man-ifold119872 onto an almost paracontact pseudometric manifold119872and let the fibres of 119891 be pseudo-Riemannian submanifoldsof 119872 Then the submersion is an affine map if and only ifℎ(nabla119864ℎ119865)+A

ℎ119864V119865+TV119864V119865 is 119891-related to nabla

119883119884 for any 119864 119865 isin

Γ(119879119872)

Proof For any 119864 119865 isin Γ(119879119872) with 119891lowastℎ119864 = 119883 ∘ 119891 and 119891

lowastV119865 =

119884 ∘ 119891 we have

(nabla119891lowast) (119864 119865) = (nabla

119891lowastℎ119864(119891lowastℎ119865)) ∘ 119891 minus 119891

lowast(ℎ (nabla119864119865))

= (nabla119883119884) ∘ 119891 minus 119891

lowast(ℎ (nablaℎ119864ℎ119865 + nabla

ℎ119864V119865

+nablaV119864ℎ119865 + nablaV119864ℎ119865))

(91)

By using (27) and (31) in the above equation we have

(nabla119891lowast) (119864 119865) = (nabla

119883119884) ∘ 119891 minus 119891

lowast(ℎ (nabla119864ℎ119865) +A

ℎ119864V119865

+TV119864V119865) (92)

Let the submersion map be affine Then for any 119864 119865 isin

Γ(119879119872) (nabla119891lowast)(119864 119865) = 0 Equation (92) implies (nabla

119883119884) ∘ 119891 =

119891lowast(ℎ(nabla119864ℎ119865) +A

ℎ119864V119865 +TV119864ℎ119865)

Conversely let ℎ(nabla119864119865) + A

ℎ119864V119865 + TV119864ℎ119865 be 119891-related

to nabla119883119884 for any 119864 119865 isin Γ(119879119872) Then from (92) we have

(nabla119891lowast)(119864 119865) = 0Hence the submersion map 119891 is affine

4 Curvature Properties

In this section the paraholomorphic bisectional curvaturesand paraholomorphic sectional curvatures of total mani-fold base manifold and fibres of paracomplex paracontactpseudo-Riemannian submersion and their curvature proper-ties are studied

Let 119891 119872 rarr 119872 be a paracomplex paracontact pseudo-Riemannian submersion from an almost para-Hermitianmanifold (119872 119869 119892) onto an almost paracontact pseudometricmanifold (119872 120601 120585 120578 119892)

Suppose that the vector fields 119864 119865 span the 2-dimensional plane at point 119901 of 119872 and let R be theRiemannian curvature tensor of 119872 The paraholomorphicbisectional curvature 119861(119864 119865) of 119872 for any pair of nonzeronon-lightlike vector fields 119864 119865 on 119872 is defined by theformula

119861 (119864 119865) =

R (119864 119869119864 119865 119869119865)

119892 (119864 119864) 119892 (119865 119865) (93)

For a nonzero non-lightlike vector field 119864 the vector field119869119864 is also non-lightlike and 119864 119869119864 span the 2-dimensionalplane Then the paraholomorphic sectional curvature 119867(119864)is defined as

119867(119864) = 119861 (119864 119864) =

R (119864 119869119864 119864 119869119864)

119892 (119864 119864) 119892 (119864 119864) (94)

The curvature properties of Riemannian submersion andsemi-Riemannian submersion have been extensively studiedin the work of OrsquoNeill [1] and Gray [3]

Let 119861ℎ and 119861V be the paraholomorphic bisectional cur-vatures of horizontal and vertical spaces respectively Let119867ℎ and 119867V be the paraholomorphic sectional curvatures of

horizontal and vertical spaces respectively Let 119861 and 119867 bethe paraholomorphic bisectional and sectional curvatures ofthe base manifold respectively

Proposition 25 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion from an almostpara-Hermitian manifold onto an almost paracontact pseudo-metric manifold and let the fibres of 119891 be pseudo-Riemanniansubmanifolds of 119872 Let 119880 119881 be non-lightlike unit verticalvector fields and let119883 119884 be non-lightlike unit horizontal vectorfields on119872 Then one has

119861 (119880119881) = 119861V(119880 119881) + 119892 (T

119880(119869119881) T

119869119880119881)

minus 119892 (T119869119880(119869119881) T

119880119881)

(95)

Geometry 11

119861 (119883119880) = 119892 ((nabla119880A)119883

119869119883 119869119880) minus 119892 (A119883119869119880A

119869119883119880)

+ 119892 (A119883119880A119869119883119869119880) minus 119892 ((nabla

119869119880A)119883

119869119883119880)

+ 119892 (T119869119880119883T119880(119869119883))

minus 119892 (T119880119883T119869119880(119869119883))

(96)

119861 (119883 119884) = 119861ℎ

(119883 119884) minus 2119892 (A119883(119869119883) A

119884(119869 119884))

+ 119892 (A119869119883119884A119883(119869 119884))

minus 119892 (A119883119884A119869119883(119869 119884))

(97)

Proof Using Definitions (93) and (94) of paraholomorphicsectional curvature and fundamental equations of submer-sion obtained by OrsquoNeill [1] we have (95) (96) and (97)

Corollary 26 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion from an almostpara-Hermitian manifold onto an almost paracontact pseudo-metric manifold If the fibres of 119891 are totally geodesic pseudo-Riemannian submanifolds of119872 then for any non-lightlike unitvertical vector fields 119880 and 119881 one has

119861 (119880119881) = 119861V(119880 119881) (98)

Corollary 27 Let 119891 119872 rarr 119872 be a paracomplex para-contact pseudo-Riemannian submersion from an almost para-Hermitian manifold onto an almost paracontact pseudometricmanifold and let the fibres of 119891 be totally geodesic pseudo-Riemannian submanifolds of119872 If the horizontal distributionis integrable then for any non-lightlike unit horizontal vectorfields119883 and 119884 one has

119861 (119883 119884) = 119861ℎ

(119883 119884) (99)

Proposition 28 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion from an almostpara-Hermitian manifold onto an almost paracontact pseudo-metric manifold and let the fibres of 119891 be pseudo-Riemanniansubmanifolds of119872 Let 119880 and 119883 be non-lightlike unit verticalvector field and non-lightlike unit horizontal vector fieldrespectively Then one has

119867(119880) = 119867V(119880) +

10038171003817100381710038171003817T119880(119869119880)

10038171003817100381710038171003817

2

minus 119892 (T119869119880(119869119880) T

119880119880)

(100)

119867(119883) = 119867 (119883) ∘ 119891 minus 310038171003817100381710038171003817A119883(119869119883)

10038171003817100381710038171003817

2

(101)

Proof The proof is straightforward If we take 119880 = 119881 in (95)and119883 = 119884 in (97) we have (98) and (99)

Corollary 29 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion from an almostpara-Hermitian manifold onto an almost paracontact pseudo-metric manifold If the fibres of 119891 are totally geodesic pseudo-Riemannian submanifolds of 119872 then the total manifold andfibres of119891have the same paraholomorphic sectional curvatures

Proof Since the fibres are totally geodesic T = 0 conse-quently we have

119867(119880) = 119867V(119880) (102)

Corollary 30 Let 119891 119872 rarr 119872 be a paracomplex para-contact pseudo-Riemannian submersion from an almost para-Hermitian manifold onto an almost paracontact pseudometricmanifold and let the fibres of 119891 be totally geodesic pseudo-Riemannian submanifolds of119872 If the horizontal distributionis integrable then the base manifold and horizontal distribu-tion have the same paraholomorphic sectional curvatures

Proof Since the horizontal distribution is integrableA = 0consequently we have

119867(119883) = 119867 (119883) ∘ 119891 (103)

Theorem 31 Let 119891 119872119898

rarr 119872119899 be a paracomplex

paracontact pseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometricmanifold 119872 and let the fibres of 119891 be pseudo-Riemanniansubmanifolds of 119872 If 119880 119881 are the non-lightlike unit verticalvector fields and 119883 119884 are the non-lightlike unit horizontalvector fields then one has

119861 (119880119881) = 119861V(119880 119881) (104)

119861 (119883119880) = minus210038171003817100381710038171003817T11988011988310038171003817100381710038171003817

2

(105)

119861 (119883 119884) = 119861 (119883 119884) ∘ 119891 (106)

Proof Using results of Lemma 19 in (95) we have

119861 (119880119881) = 119861V(119880 119881) minus 119892 (119869 (T

119880119881) 119869 (T

119880119881))

minus 119892 (1198692

(T119880119881) T

119880119881)

= 119861 (119880119881) + 119892 (T119880119881T119880119881) minus 119892 (T

119880119881T119880119881)

= 119861V(119880 119881)

(107)

Applying results of Lemma 19 in (96) we have

119861 (119883119880) = 119892 ((nabla119880A)119883

(119869119883) 119869119880) minus 119892 ((nabla119869119880A)119883

(119869119883) 119880)

+ 21003817100381710038171003817A119883119880

1003817100381710038171003817

2

minus 210038171003817100381710038171003817T11988011988310038171003817100381710038171003817

2

(108)

Since byTheorem 21 the horizontal distribution is integrablewe haveA = 0 which implies

119861 (119883119880) = minus210038171003817100381710038171003817T11988011988310038171003817100381710038171003817

2

(109)

In view ofA = 0 (104) follows from (97)

12 Geometry

Theorem 32 Let 119891 119872119898

rarr 119872119899 be a paracomplex

paracontact pseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometricmanifold 119872 and let the fibres of 119891 be pseudo-Riemanniansubmanifolds of119872 If 119880 119883 are non-lightlike unit vertical andnon-lightlike unit horizontal vector fields respectively then onehas

119867(119880) = 119867V(119880) minus 2

1003817100381710038171003817T1198801198801003817100381710038171003817

2

(110)

119867(119883) = 119867 (119883) ∘ 119891 (111)

Proof Since 119891 is the paracomplex paracontact pseudo-Riemannian submersion from a para-Kahler manifold 119872

onto an almost paracontact pseudometric manifold 119872 by(16) and equations of Lemma 19 we have

119892 (T119869119880(119869119880) T

119880119880) = 119892 (119869

2

(T119880119880) T

119880119880) =

1003817100381710038171003817T1198801198801003817100381710038171003817

2

119892 (T119880(119869119880) T

119880(119869119880)) = minus119892 (T

119880119880T119880119880) = minus

1003817100381710038171003817T1198801198801003817100381710038171003817

2

(112)

and by using the above results in (100) we have

119867(119880) = 119867V(119880) minus

1003817100381710038171003817T1198801198801003817100381710038171003817

2

minus1003817100381710038171003817T119880119880

1003817100381710038171003817

2

= 119867V(119880) minus 2

1003817100381710038171003817T1198801198801003817100381710038171003817

2

(113)

Again since horizontal distribution is integrable we haveA = 0 and putting it in (101) we obtain (111)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Uma Shankar Verma is thankful to University Grant Com-mission New Delhi India for financial support

References

[1] B OrsquoNeill ldquoThe fundamental equations of a submersionrdquo TheMichigan Mathematical Journal vol 13 pp 459ndash469 1966

[2] B OrsquoNeill Semi-Riemannian Geometry with Applications toRelativity vol 103 of Pure and Applied Mathematics AcademicPress New York NY USA 1983

[3] A Gray ldquoPseudo-Riemannian almost product manifolds andsubmersionsrdquo vol 16 pp 715ndash737 1967

[4] J P Bourguignon and H B Lawson A Mathematicians Visitto Kaluza-Klein Theory Rendiconti del Seminario Matematico1989

[5] S Ianus and M Visinescu ldquoSpace-time compactification andRiemannian submersionsrdquo in The Mathematical Heritage GRassias and C F Gauss Eds pp 358ndash371 World ScientificRiver Edge NJ USA 1991

[6] J P Bourguignon and H B Lawson Jr ldquoStability and isolationphenomena for Yang-Mills fieldsrdquo Communications in Mathe-matical Physics vol 79 no 2 pp 189ndash230 1981

[7] B Watson ldquoG 1198661015840

-Riemannian submersions and nonlineargauge field equations of general relativityrdquo in Global AnalysismdashAnalysis on Manifolds T Rassias and M Morse Eds vol 57 ofTeubner-Texte zur Mathematik pp 324ndash349 Teubner LeipzigGermany 1983

[8] C Altafini ldquoRedundant robotic chains on Riemannian submer-sionsrdquo IEEE Transactions on Robotics and Automation vol 20no 2 pp 335ndash340 2004

[9] M T Mustafa ldquoApplications of harmonic morphisms to grav-ityrdquo Journal of Mathematical Physics vol 41 no 10 pp 6918ndash6929 2000

[10] B Watson ldquoAlmost Hermitian submersionsrdquo Journal of Differ-ential Geometry vol 11 no 1 pp 147ndash165 1976

[11] D Chinea ldquoAlmost contact metric submersionsrdquo Rendiconti delCircolo Matematico di Palermo II vol 34 no 1 pp 319ndash3301984

[12] D Chinea ldquoTransference of structures on almost complexcontact metric submersionsrdquo Houston Journal of Mathematicsvol 14 no 1 pp 9ndash22 1988

[13] Y Gunduzalp and B Sahin ldquoPara-contact para-complexpseudo-Riemannian submersionsrdquo Bulletin of the MalaysianMathematical Sciences Society

[14] I Sato ldquoOn a structure similar to the almost contact structurerdquoTensor vol 30 no 3 pp 219ndash224 1976

[15] P K Rashevskij ldquoThe scalar field in a stratified spacerdquoTrudy Seminara po Vektornomu i Tenzornomu Analizu s ikhPrilozheniyami k Geometrii Mekhanike i Fizike vol 6 pp 225ndash248 1948

[16] P Libermann ldquoSur les structures presque paracomplexesrdquoComptes Rendus de lrsquoAcademie des Sciences I vol 234 pp 2517ndash2519 1952

[17] E M Patterson ldquoRiemann extensions which have Kahlermetricsrdquo Proceedings of the Royal Society of Edinburgh AMathematics vol 64 pp 113ndash126 1954

[18] S Sasaki ldquoOn differentiable manifolds with certain structureswhich are closely related to almost contact structure Irdquo TheTohoku Mathematical Journal vol 12 pp 459ndash476 1960

[19] S Zamkovoy ldquoCanonical connections on paracontact mani-foldsrdquo Annals of Global Analysis and Geometry vol 36 no 1pp 37ndash60 2009

[20] D E Blair Riemannian Geometry of Contact and SymplecticManifolds vol 23 of Progress in Mathematics BirkhauserBoston Mass USA 2002

[21] V Cruceanu P Fortuny and P M Gadea ldquoA survey onparacomplex geometryrdquoThe Rocky Mountain Journal of Mathe-matics vol 26 no 1 pp 83ndash115 1996

[22] P M Gadea and J M Masque ldquoClassification of almost para-Hermitian manifoldsrdquo Rendiconti di Matematica e delle sueApplicazioni VII vol 7 no 11 pp 377ndash396 1991

[23] M Falcitelli S Ianus and A M Pastore Riemannian Submer-sions and Related Topics World Scientific River Edge NJ USA2004

[24] E G Rio and D N Kupeli Semi-Riemannian Maps andTheir Applications Kluwer Academic Publisher DordrechtTheNetherlands 1999

[25] B Sahin ldquoSemi-invariant submersions from almost HermitianmanifoldsrdquoCanadianMathematical Bulletin vol 54 no 3 2011

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 5: Research Article Paracomplex Paracontact Pseudo-Riemannian …downloads.hindawi.com/archive/2014/616487.pdf · 2019-07-31 · Research Article Paracomplex Paracontact Pseudo-Riemannian

Geometry 5

On operating 119869 on both sides of the above equation weget

119880 = 120601V(120601

V(119880)) + 120578

V(120601

V(119880)) 120585

ℎ

+ 120578V(119880) 120585

V (47)

where 119869(120585ℎ

) = 120585V

Equating horizontal and vertical components we have

119880 = 120601V(120601

V(119880)) + 120578

V(119880) 120585

V 120578

V∘ 120601

V(119880) = 0

997904rArr (120601V)2

(119880) = 119880 minus 120578V(119880) 120585

V 120578

V∘ 120601

V= 0

120601V(120585

V) = 0 120578

V(120585

V) = 1

(48)

Hence (120601V 120585

V 120578

V 119892

V) is almost paracontact pseudometric

structure on the fibre 119891minus1(119902) 119902 isin 119872

Proposition 6 Let 119891 1198722119898

rarr 1198722119899+1 be a paracomplex

paracontact pseudo-Riemannian submersion and let the fibresof 119891 be pseudo-Riemannian submanifolds of 119872 Let 119883 119884 bebasic vector fields 119891-related to 119883 119884 respectively Let 120578 and 120578be 1-forms on the total manifold119872 and the base manifold119872respectively Then one has the following

(i) The characteristic vector field 119869120585ℎ

is a vertical vectorfield

(ii) 119891lowastlowast120578 = 120578ℎ where 119891lowast

lowast120578 is pullback of 120578 through 119891

lowast

(iii) 120578ℎ(119880) = 0 for any vertical vector field 119880(iv) 120578V(119883) = 0 for any horizontal vector field119883

Remark 7 Results (ii) and (iv) are analogue version of results(i) and (iii) of Proposition 4 of [13]

Proof (i) By Corollary 5 (120601V 120585

V 120578

V 119892

V) is almost paracontact

pseudometric structure on 119891minus1(119902) We have

0 = 119892 (120585ℎ

119869120585ℎ

) = 119892(119891lowast(120585ℎ

) 119891lowast(119869120585ℎ

))

= 119892(120585 119891lowast(119869120585ℎ

))

(49)

Now

119891lowast(119869120585ℎ

) = 120601 ∘ 119891lowast120585ℎ

+ 120578 (120585ℎ

) 120585 = 120578 (120585ℎ

) 120585 (50)

so we have

0 = 119892(120585 120578 (120585ℎ

) 120585) = 120578 (120585ℎ

)119892 (120585 120585) = 120578 (120585ℎ

) (51)

Thus 119891lowast(119869120585ℎ

) = 0Hence 119869120585

ℎ

is a vertical vector field(ii) Since 119891 119872

2119898

rarr 1198722119899+1 is smooth submersion

120578ℎ

= 120578|H

is restriction of 120578 on the horizontal distribution

H and 119891lowast119901

H119901rarr 119879119891(119901)

119872 is a linear isometry for any119883119901isinH119901 we get

120578ℎ

119901(119883119901) = 120576119892

119901(120585ℎ

119901 119883119901) = 119892

119891(119901)(119891lowast119901120585ℎ

119901 119891lowast119901119883119901)

= 119892119891(119901)

(120585119891(119901)

119883119891(119901)

) = 120578119891(119901)

(119883119891(119901)

) = 119891lowast

lowast120578119901(119883119901)

(52)

Hence pullback 119891lowastlowast120578 = 120578ℎ

Results (iii) and (iv) immediately follow from the previ-ous results

Example 8 Let (R42 119869 119892) be a paracomplex pseudometric

manifold and let (R31 120601 120585 120578 119892) be an almost paracontact

pseudometric manifoldDefine a submersion 119891 R4

2 (1199091 1199092 1199101 1199102)119905

rarr

R31 (119906 V 119908)119905 by

119891 ((1199091 1199092 1199101 1199102)119905

) 997891997888rarr (1199091+ 1199092+ 31199101+ 21199102

31199091+ 21199092+ 1199101+ 1199102

51199091+ 31199092+ 51199101+ 31199102)119905

(53)

Then the kernel of 119891lowastis

V = ker119891lowast= Span119881

1=

120597

1205971199091

minus 2120597

1205971199092

minus120597

1205971199101

+ 2120597

1205971199102

(54)

which is the vertical distribution admitting one lightlikevector field that is fibre is degenerate submanifold of R4

2

The horizontal distribution is

H = (ker119891lowast)perp

= Span1198831=

120597

1205971199091

minus120597

1205971199101

1198832=

120597

1205971199092

+ 2120597

1205971199101

1198833= 2

120597

1205971199101

+120597

1205971199102

(55)

For any real 119896 the horizontal characteristic vector field 120585ℎ

isgiven by

120585ℎ

= 119896120597

1205971199091

minus (2119896 minus1

3)

120597

1205971199092

minus (119896 minus 1)120597

1205971199101

+ (2119896 minus5

3)

120597

1205971199102

(56)

which is 119891-related to the characteristic vector field 120585 = 120597120597119908Moreover there exists one form 120578 = 5119889119909

1+3119889119909

2+51198891199101+

31198891199102on (R4

2 119869 119892) such that the submersion satisfies (39)

Example 9 Let (R63 119869 119892) be an almost paracomplex pseudo-

Riemannian manifold and let (R31 120601 120585 120578 119892) be an almost

6 Geometry

paracontact pseudo-Riemannian manifold Consider a sub-mersion 119891 R6

3 (1199091 1199092 1199093 1199101 1199102 1199103)119905

rarr R31 (119906 V 119908)119905

defined by

119891 ((1199091 1199092 1199093 1199101 1199102 1199103)119905

)

997891997888rarr (1199091+ 1199092

radic2

1199101+ 1199102

radic2

1199102+ 1199103

radic2

)

119905

(57)

Then there exists one form 120578 = (1198891199092+ 1198891199093)radic2 on (R6

3 119869 119892)

such that (39) is satisfied The kernel of 119891lowastis

V = ker119891lowast

= Span1198811=

120597

1205971199091

minus120597

1205971199092

1198812=

120597

1205971199101

minus120597

1205971199102

+120597

1205971199103

1198813=

120597

1205971199093

(58)

which is vertical distribution admitting non-lightlike vectorfields that is the fibre is nondegenerate submanifold of(R63 119869 119892)The horizontal distribution is

H = Span1198831=

120597

1205971199091

+120597

1205971199092

1198832= minus

120597

1205971199101

+120597

1205971199103

1198833=

120597

1205971199101

+120597

1205971199102

(59)

Example 10 Let (R42 119869 119892) be a paracomplex pseudometric

manifold and let (R31 120601 120585 120578 119892) be an almost paracontact

pseudometric manifoldConsider a submersion 119891 R4

2 (1199091 1199092 1199101 1199102)119905

rarr R31

(119906 V 119908)119905 defined by

119891 ((1199091 1199092 1199101 1199102)119905

) 997891997888rarr (1199091 1199101 1199102)119905

(60)

Then the kernel of 119891lowastis

V = ker119891lowast= Span119881

1=

120597

1205971199092

(61)

which is the vertical distribution and the restriction of 119892 tothe fibres of 119891 is nondegenerate

The horizontal distribution is

H = (ker119891lowast)perp

= Span119883 =120597

1205971199091

119884 =120597

1205971199101

120585ℎ

=120597

1205971199102

(62)

The characteristic vector field 120585 = 120597120597119908 on R31has unique

horizontal lift 120585ℎ

which is the characteristic vector field onhorizontal distributionH of R4

2

We also have

119892 (119883119883) = 119892 (119891lowast119883119891lowast119883) = minus1

119892 (119884 119884) = 119892 (119891lowast119884 119891lowast119884) = 1

119892 (120585ℎ

120585ℎ

) = 119892(119891lowast120585ℎ

119891lowast120585ℎ

) = 119892 (120585 120585) = 1

(63)

Thus the smooth map 119891 is a pseudo-Riemannian submer-sion

Moreover we obtain that there exists a 1-form 120578 = 1198891199092on

R42such that 120578(119869120585

ℎ

) = 1 120578(120585ℎ

) = 0 and the map 119891 satisfies

119891lowast119869119883 = 120601119891

lowast119883 + 120578 (119883) 120585

119891lowast119869119884 = 120601119891

lowast119884 + 120578 (119884) 120585

119891lowast119869120585ℎ

= 120601119891lowast120585ℎ

+ 120578 (120585ℎ

) 120585

(64)

Hence the map 119891 is a paracomplex paracontact pseudo-Ri-emannian submersion from R4

2on to R3

1

Proposition 11 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion and let the fibresof 119891 be pseudo-Riemannian submanifolds of 119872 Let 119883 119884 bebasic vector fields 119891-related to119883 119884 respectively Then 119869(119883) minus120576119892(119883 120585

ℎ

)120585ℎ

is 119891-related to 120601119883

Proof Since119883 is 119891-related to vector field119883 on119872 we have

120578 (119883) = 120578V+ 120578ℎ

(119883) = 0 + 120578ℎ

(119883) = 120576119892 (119883 120585ℎ

)

997904rArr 119891lowast(119869119883) = 120601119883 + 120578

ℎ

(119883) 120585

997904rArr 119891lowast119869119883 minus 120576119892 (119883 120585

ℎ

) 120585ℎ

= 120601119883

(65)

Hence 119869(119883) minus 120576119892(119883 120585ℎ

)120585ℎ

is 119891-related to 120601119883

Proposition 12 Let 119891 119872 rarr 119872 be a paracomplex para-contact pseudo-Riemannian submersion and let the fibres of 119891be pseudo-Riemannian submanifolds of 119872 Let V and H bethe vertical and horizontal distributions respectively If 120585

ℎ

isthe basic characteristic vector field of horizontal distribution119891-related to the characteristic vector field 120585 of base manifoldthen

(i) 119869V sub DVoplus 119869120585ℎ

oplus 120585ℎ

(ii) 119869H sub Dℎ

oplus 120585ℎ

oplus 119869120585ℎ

Proof (i) Let 119880 isin V Then 119880 = 119886119880|DV + 119887119869120585

ℎ

for 119886 119887 isin

119862infin

(119872) as 119869120585ℎ

= 120585Vis characteristic vector field on odd

Geometry 7

dimensional fibre submanifold 119891minus1(119902) of119872 119902 isin 119872 We get

119869119880 = 119886119869119880|DV + 119887119869

2

120585ℎ

= 119886119869119880|DV + 119887120585

ℎ

isinV oplus 120585ℎ

997904rArr 119869V subV oplus 120585ℎ

(66)

Again let 119881 isinV oplus 120585ℎ

Then 119881 = 119886119881|DV + 119887119869120585

ℎ

+ 119888120585ℎ

where

120578V(119881|DV ) = 0 D

V= ker 120578V 119886119881

|DV + 119887119869120585

ℎ

isin V and 119886 119887 119888 isin119862infin

(119872) We have

119869119881 = 119886119869119881|DV + 119887120585

ℎ

+ 119888119869120585ℎ

= (119886119869119881|DV + 119888119869120585

ℎ

)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

isinV

+ 119887120585ℎ

⏟⏟⏟⏟⏟⏟⏟

isin120585

ℎ

isinV oplus 120585ℎ

(67)

Now by (39) we get

119891lowast119869119881 = 120601 (119891

lowast119881) + 120578 (119891

lowast(119881)) 120585

= 119888 120601 (119891lowast120585ℎ

) + 120578 (120585) 120585

= 119888120585 isin 120585 sube 119869V

(68)

We get 119869119881 notinVHence 119869V subV oplus 120585

ℎ

that is 119869V sub DVoplus 119869120585ℎ

oplus 120585ℎ

(ii) Let 119883 = 119886119883

|

Dℎ+ 119887120585ℎ

isin H where H = Dℎ

oplus

120585ℎ

ker 120578ℎ = Dℎ

and 119886 119887 isin 119862infin(119872) Then

119869119883 = 119886119869119883|

Dℎ+ 119887119869120585ℎ

isinH oplus 119869120585ℎ

(69)

which implies that 119869H subH oplus 119869120585ℎ

Again let119884 isinHoplus119869120585

ℎ

Then119884 = 119886119884|

Dℎ+119887120585ℎ

+119888119869120585ℎ

notinHfor 119886 119887 119888 isin 119862infin(119872) We have

119869 119884 = 119886119869119884|

Dℎ+ 119887119869120585ℎ

+ 1198881198692

120585ℎ

= 119886119869119884|

Dℎ+ 119887119869120585ℎ

+ 119888120585ℎ

= 119885 + 119887119869120585ℎ

isinH oplus 119869120585ℎ

for some 119885 = 119886119869119884|

Dℎ+ 119888120585ℎ

isinH

(70)

We obtain 119869 119884 notinHHence 119869H sub H oplus 119869(120585

ℎ

) that is 119869H sub Dℎ

oplus 120585ℎ

oplus

119869120585ℎ

Example 13 Let (R63 119869 119892) be an almost paracomplex pseudo-

Riemannian manifold and let (R31 120601 120585 120578 119892) be an almost

paracontact pseudo-Riemannian manifold Consider a sub-mersion 119891 R6

3 (1199091 1199092 1199093 1199101 1199102 1199103)119905

rarr R31 (119906 V 119908)119905

defined by

119891 ((1199091 1199092 1199093 1199101 1199102 1199103)119905

) 997891997888rarr (1199091+ 1199092

radic2

1199101+ 1199102

radic2

1199103)

119905

(71)

Then the kernel of 119891lowastis

V = ker119891lowast

= Span1198811=

120597

1205971199091

minus120597

1205971199092

1198812=

120597

1205971199101

minus120597

1205971199102

120585V=

120597

1205971199093

(72)

which is the vertical distribution and the restriction of 119892 tothe fibres of 119891 is nondegenerate

The horizontal distribution is

H = (ker119891lowast)perp

= Span1198831=

120597

1205971199091

+120597

1205971199092

1198832=

120597

1205971199101

+120597

1205971199102

120585ℎ

=120597

1205971199103

(73)

The characteristic vector field 120585 = 120597120597119908 on R31has unique

horizontal lift 120585ℎ

which is the characteristic vector field onthe horizontal distributionH of R6

3

We also have

119892 (1198831 1198831) = 119892 (119891

lowast1198831 119891lowast1198831) = minus2

119892 (1198832 1198832) = 119892 (119891

lowast1198832 119891lowast1198832) = 2

119892 (120585ℎ

120585ℎ

) = 119892(119891lowast120585ℎ

119891lowast120585ℎ

) = 119892 (120585 120585) = 1

(74)

Thus the smooth map 119891 is a pseudo-Riemannian submer-sion

Also we obtain that there exists a 1-form 120578 = 1198891199093on R63

such that 120578(119869120585ℎ

) = 1 120578(120585ℎ

) = 0 and the map 119891 satisfies

119891lowast1198691198831= 120601119891lowast1198831+ 120578 (119883

1) 120585

119891lowast1198691198832= 120601119891lowast1198832+ 120578 (119883

2) 120585

119891lowast119869120585ℎ

= 120601119891lowast120585ℎ

+ 120578 (120585ℎ

) 120585

(75)

Hence the map 119891 is a paracomplex paracontact pseudo-Riemannian submersion from R6

3onto R3

1

Moreover we observe that for this submersion119891 we have

119869V subV oplus 120585ℎ

119869H subH oplus 119869120585ℎ

(76)

which verifies Proposition 12

8 Geometry

Proposition 14 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion and let the fibresof 119891 be pseudo-Riemannian submanifolds of 119872 Let 119883 119884 bebasic vector fields 119891-related to 119883 119884 respectively Let 119865 and Φbe the second fundamental forms and let nabla and nabla be the Levi-Civita connection on the total manifold119872 and base manifold119872 respectively Then one has

(i) 119891lowast((nabla119883119869)119884) = (nabla

119883120601)119884 + 120576119892(119884 nabla

119883120585)120585 + 120578(119884)nabla

119883120585

(ii) 119865 = 119891lowastlowastΦ + 120576120578 otimes 120578

(iii) 119891lowast((nabla119883119865)(119884 119885)) = (nabla

119883Φ)(119884 119885) + 120578(119884)119892(119885 nabla

119883120585) +

120578(119885)119892(119884 nabla119883120585)

Proof (i) In view of Definition 2 and Proposition 11 we have

119891lowast((nabla119883119869) 119884) = 119891

lowast(nabla119883(119869 119884) minus 119869 (nabla

119883119884))

= nabla119883(119891lowast(119869 119884)) minus 119891

lowast(119869 (nabla119883119884))

= nabla119883(120601119884) + nabla

119883(120578 (119884) 120585) minus 120601 (nabla

119883119884)

minus 120578 (nabla119883119884) 120585

= (nabla119883120601)119884 + nabla

119883(120576119892 (119884 120585) 120585) minus 120578 (nabla

119883119884) 120585

= (nabla119883120601)119884 + 120576119892 (nabla

119883119884 120585) 120585 + 120576119892 (119884 nabla

119883120585) 120585

+ 120576119892 (119884 120585) nabla119883120585 minus 120576119892 (nabla

119883119884 120585) 120585

= (nabla119883120601)119884 + 120576119892 (119884 nabla

119883120585) 120585 + 120578 (119884) nabla

119883120585

(77)

(ii) Since119891lowastlowastΦ is pullback ofΦ through the linearmap119891

lowast

we get

119891lowast

lowastΦ(119883 119884) = Φ (119883 119884) ∘ 119891 = 119892 (119883 120601119884) ∘ 119891

= 119892 (119883 119869 119884) minus 120576120578 (119884) 120578 (119883)

= 119865 (119883 119884) minus 120576120578 (119883) 120578 (119884)

(78)

which implies 119865 = 119891lowastlowastΦ + 120576120578 otimes 120578

(iii) By (23) we have

119891lowast((nabla119883119865) (119884 119885)) = 119892 (119891

lowast(119884) 119891

lowast((nabla119883119869)119885)) (79)

Now using (i) in the above equation we get (iii)

Theorem 15 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion and let the fibresof 119891 be pseudo-Riemannian submanifolds of 119872 Let 119883 119884 bebasic vector fields 119891-related to 119883 119884 respectively If the totalspace is para-Hermitian manifold then the almost paracontactstructure of base space is normal

Moreover if the almost paracontact structure of base spaceis normal then the Nijenhuis tensor of total space is vertical

Proof The Nijenhuis tensors 119873119869and 119873

120601of almost para-

complex structure 119869 and almost paracontact structure 120601 arerespectively defined by (8) and (11)

Using Definition 2 and properties of Sections 21 and 22we get the following identity

119891lowast(119873119869(119883 119884)) = 119873

(1)

(119883 119884) + 2119889120578 (120601119883 119884) 120585

minus 2119889120578 (120601119884119883) 120585

+ 2120578 (119883) 119889120578 (120585 119884) 120585 minus 2120578 (119884) 119889120578 (120585 119883) 120585

minus 120578 (119884)119873(3)

(119883) + 120578 (119883)119873(3)

(119884)

(80)

Using (12) (13) (14) and (15) (80) reduces to

119891lowast(119873119869(119883 119884)) = 119873

(1)

(119883 119884) + 119873(2)

(119883 119884) 120585

+ 120578 (119883)119873(4)

(119884) 120585

minus 120578 (119884)119873(4)

(119883) 120585 minus 120578 (119884)119873(3)

(119883)

+ 120578 (119883)119873(3)

(119884)

(81)

Since 119873119869(119883 119884) = 0 it follows from (81) that

tensors 119873(1) 119873(2) 119873(3) and 119873(4) vanish togetherHence the almost paracontact structure of base space is

normalConversely let the almost paracontact structure of the

base space be normalThen (81) implies that 119891

lowast(119873119869(119883 119884)) = 0

Hence119873119869(119883 119884) is vertical

Corollary 16 Let119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion and let the fibres of 119891 bepseudo-Riemannian submanifolds of 119872 Let 119883 119884 be basicvector fields 119891-related to 119883 119884 respectively Let the total spacebe para-Hermitian manifold and119873(1) vanishes Then the basespace is paracontact pseudometric manifold if and only if 120585 iskilling

Proof Let the total space be para-Hermitian and 119873(1) van-ishes Then from (80) we have

0 = 2119889120578 (120601119883 119884) 120585 minus 2119889120578 (120601119884119883) 120585 + 2120578 (119883) 119889120578 (120585 119884) 120585

minus 2120578 (119884) 119889120578 (120585 119883) 120585 minus 120578 (119884) (L120585120601)119883 + 120578 (119883) (L

120585120601)119884

(82)

If 120585 is killing then we have L120585120601 = 0 It immediately follows

from (82) that

119889120578 (120601119883 119884) minus 120578 (120601119884119883) + 120578 (119883) 119889120578 (120585 119884)

minus 120578 (119884) 119889120578 (120585 119883) = 0

(83)

In view of (6) and (7) the above equation gives 119889120578 = ΦConversely let the base space be paracontact Then 119889120578 =

ΦUsing (6) (7) and (82) we getL

120585120601 = 0

Hence the characteristic vector field 120585 is killing

Geometry 9

Theorem 17 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion and let the fibres of 119891 bepseudo-Riemannian submanifolds of 119872 Let 119883 119884 be basicvector fields 119891-related to 119883 119884 respectively If the total spaceis para-Kahler then the base space is paracosymplectic Theconverse is true if nabla

119883119869 is vertical

Proof We have for any 119883119884 isin Γ(119879119872) (nabla119883120601)119884 = 0 which

gives 119892(119885 (nabla119883120601)119884) = 0 for any 119885 isin Γ(119879119872)

From Proposition 14 we have

119892 (119885 (nabla119883120601)119884) + 120576120578 (119884) (nabla

119883120578)119885 + 120576120578 (119885) (nabla

119883120578) 119884 ∘ 119891

= 119892 (119885 (nabla119883119869) 119884)

(84)

Let nabla 119869 = 0 that is the total space is para-KahlerThen from(84) we obtain nabla120601 = 0 and nabla120578 = 0 Hence the base space isparacosymplectic

Again let (nabla119883120601)119884 = 0 and nabla120578 = 0 Then 119892(119885 (nabla

119883119869)119884) =

0 which implies that (nabla119883119869)119884 is a vertical vector field

Theorem 18 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion and let the fibres of 119891 bepseudo-Riemannian submanifolds of 119872 Let 119883 119884 and 119885 bebasic vector fields 119891-related to 119883 119884 and 119885 respectively If120578(119883)119885(120578(119884))+120578(119884)119883(120578(119885))minus120578(119883)119884(120578(119885)) = 0 then the totalspace is almost para-Kahler if and only if the base space119872 isan almost paracosymplectic manifold

Proof We have the following equation

3119889119865 (119883 119884 119885)

= 3 (119891lowast

lowast119889Φ) (119883 119884 119885) + 2120576120578 (119885) 119889120578 (119883 119884)

minus 2120576120578 (119884) 119889120578 (119883 119885)

+ 2120576120578 (119883) 119889120578 (119884 119885) + 2120576120578 (119883)119885 (120578 (119884))

+ 2120576120578 (119884)119883 (120578 (119885))

minus 2120576120578 (119883)119884 (120578 (119885))

(85)

If 119889120578 = 0 119889Φ = 0 and 120578(119883)119885(120578(119884)) + 120578(119884)119883(120578(119885)) minus

120578(119883)119884(120578(119885)) = 0 then from (85) we have 119889119865 = 0 Hencethe total space is almost para-Kahler

Conversely let 119889119865 = 0 and 120578(119883)119885(120578(119884))+120578(119884)119883(120578(119885))minus120578(119883)119884(120578(119885)) = 0

By using the above equation in (85) we have 119889120578 = 0 and119889Φ = 0

Hence the base space is almost paracosymplectic

Nowwe investigate the properties of fundamental tensorsT andA of a pseudo-Riemannian submersion

Lemma 19 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometric manifold119872 and

let the fibres of 119891 be pseudo-Riemannian submanifolds of 119872Then for any horizontal vector fields119883 119884 and for any verticalvector fields 119880 119881 on119872 one has

(i) A119883(119869 119884) = 119869(A

119883119884)

(ii) A119869119883(119884) = 119869(A

119883119884)

(iii) T119880(119869 119881) = 119869(T

119880119881)

(iv) T119869119880119881 = 119869(T

119880119881)

Proof The proof follows using similar steps as in Lemmas 3and 4 of [13] so we omit it

Lemma 20 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometric manifold119872 andlet the fibres of 119891 be pseudo-Riemannian submanifolds of 119872Then for any vector fields 119864 119865 on119872 one has

(i) A119864(119869119865) = 119869(A

119864119865)

(ii) T119864(119869119865) = 119869(T

119864119865)

Proof The proof follows from (37) and (38)

Theorem 21 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometric manifold119872 andlet the fibres of 119891 be pseudo-Riemannian submanifolds of 119872Then the horizontal distribution is integrable

Proof For any vertical vector field 119880 we have

119892 (119869 (A119883119884) 119880) = 119892 (A

119883119869 119884119880)

= minus119892 (119869119884A119883119880)

= minus119892 (119869119884 ℎ (nabla119880119883))

= 119892 (119884 ℎ (119869 (nabla119880119883)))

= 119892 (119884 ℎ (minusnabla119880119869)119883

+nabla119880(119869119883))

= 119892 (119884 ℎ nabla119880(119869119883)) = 119892 (119884A

119869119883119880)

= minus119892 (A119869119883119884119880) = minus119892 (119869 (A

119883119884) 119880)

(86)

Thus 119892(119869(A119883119884) 119880) = 0 which is true for all119883 and 119884

SoA119883119884 = 0

Hence the horizontal distribution is integrable

Theorem 22 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometric manifold119872 andlet the fibres of 119891 be pseudo-Riemannian submanifolds of 119872Then the submersion is an affine map onH

10 Geometry

Proof The second fundamental form of 119891 is defined by

(nabla119891lowast) (119864 119865) = (nabla

119891

119864119891lowast(119865)) ∘ 119891 minus 119891

lowast(nabla119864119865) (87)

where 119864 119865 isin Γ(119879119872) and nabla119891 is pullback connection of Levi-Civita connection nabla on119872 with respect to 119891

We have for any119883119884 isinH

(nabla119891lowast) (119883 119884) = (nabla

119891

119883

119891lowast(119884)) ∘ 119891 minus 119891

lowast(nabla119883119884) (88)

By using Lemma 1 we have 119891lowast(ℎ(nabla119883119884)) = (nabla

119883119884) ∘ 119891 which

implies nabla119891lowast= 0

Hence the submersion 119891 is an affine map onH

Theorem 23 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Hermitian man-ifold119872 onto an almost paracontact pseudometric manifold119872and let the fibres of 119891 be pseudo-Riemannian submanifolds of119872 Then the submersion is an affine map on V if and only ifthe fibres of 119891 are totally geodesic

Proof We have for any 119880119881 isinV

(nabla119891lowast) (119880 119881) = minus119891

lowast(ℎ (nabla119880119881)) (89)

which in view of (27) gives

(nabla119891lowast) (119880 119881) = minus119891

lowast(T119880119881) (90)

Let the fibres of 119891 be totally geodesic Then T = 0Consequently from the above equation we have nabla119891

lowast= 0

Thus the map 119891 is affine onVConversely let the submersion 119891 be an affine map onV

Then nabla119891lowast= 0 which impliesT = 0

Hence the fibres of 119891 are totally geodesic

Theorem 24 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Hermitian man-ifold119872 onto an almost paracontact pseudometric manifold119872and let the fibres of 119891 be pseudo-Riemannian submanifoldsof 119872 Then the submersion is an affine map if and only ifℎ(nabla119864ℎ119865)+A

ℎ119864V119865+TV119864V119865 is 119891-related to nabla

119883119884 for any 119864 119865 isin

Γ(119879119872)

Proof For any 119864 119865 isin Γ(119879119872) with 119891lowastℎ119864 = 119883 ∘ 119891 and 119891

lowastV119865 =

119884 ∘ 119891 we have

(nabla119891lowast) (119864 119865) = (nabla

119891lowastℎ119864(119891lowastℎ119865)) ∘ 119891 minus 119891

lowast(ℎ (nabla119864119865))

= (nabla119883119884) ∘ 119891 minus 119891

lowast(ℎ (nablaℎ119864ℎ119865 + nabla

ℎ119864V119865

+nablaV119864ℎ119865 + nablaV119864ℎ119865))

(91)

By using (27) and (31) in the above equation we have

(nabla119891lowast) (119864 119865) = (nabla

119883119884) ∘ 119891 minus 119891

lowast(ℎ (nabla119864ℎ119865) +A

ℎ119864V119865

+TV119864V119865) (92)

Let the submersion map be affine Then for any 119864 119865 isin

Γ(119879119872) (nabla119891lowast)(119864 119865) = 0 Equation (92) implies (nabla

119883119884) ∘ 119891 =

119891lowast(ℎ(nabla119864ℎ119865) +A

ℎ119864V119865 +TV119864ℎ119865)

Conversely let ℎ(nabla119864119865) + A

ℎ119864V119865 + TV119864ℎ119865 be 119891-related

to nabla119883119884 for any 119864 119865 isin Γ(119879119872) Then from (92) we have

(nabla119891lowast)(119864 119865) = 0Hence the submersion map 119891 is affine

4 Curvature Properties

In this section the paraholomorphic bisectional curvaturesand paraholomorphic sectional curvatures of total mani-fold base manifold and fibres of paracomplex paracontactpseudo-Riemannian submersion and their curvature proper-ties are studied

Let 119891 119872 rarr 119872 be a paracomplex paracontact pseudo-Riemannian submersion from an almost para-Hermitianmanifold (119872 119869 119892) onto an almost paracontact pseudometricmanifold (119872 120601 120585 120578 119892)

Suppose that the vector fields 119864 119865 span the 2-dimensional plane at point 119901 of 119872 and let R be theRiemannian curvature tensor of 119872 The paraholomorphicbisectional curvature 119861(119864 119865) of 119872 for any pair of nonzeronon-lightlike vector fields 119864 119865 on 119872 is defined by theformula

119861 (119864 119865) =

R (119864 119869119864 119865 119869119865)

119892 (119864 119864) 119892 (119865 119865) (93)

For a nonzero non-lightlike vector field 119864 the vector field119869119864 is also non-lightlike and 119864 119869119864 span the 2-dimensionalplane Then the paraholomorphic sectional curvature 119867(119864)is defined as

119867(119864) = 119861 (119864 119864) =

R (119864 119869119864 119864 119869119864)

119892 (119864 119864) 119892 (119864 119864) (94)

The curvature properties of Riemannian submersion andsemi-Riemannian submersion have been extensively studiedin the work of OrsquoNeill [1] and Gray [3]

Let 119861ℎ and 119861V be the paraholomorphic bisectional cur-vatures of horizontal and vertical spaces respectively Let119867ℎ and 119867V be the paraholomorphic sectional curvatures of

horizontal and vertical spaces respectively Let 119861 and 119867 bethe paraholomorphic bisectional and sectional curvatures ofthe base manifold respectively

Proposition 25 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion from an almostpara-Hermitian manifold onto an almost paracontact pseudo-metric manifold and let the fibres of 119891 be pseudo-Riemanniansubmanifolds of 119872 Let 119880 119881 be non-lightlike unit verticalvector fields and let119883 119884 be non-lightlike unit horizontal vectorfields on119872 Then one has

119861 (119880119881) = 119861V(119880 119881) + 119892 (T

119880(119869119881) T

119869119880119881)

minus 119892 (T119869119880(119869119881) T

119880119881)

(95)

Geometry 11

119861 (119883119880) = 119892 ((nabla119880A)119883

119869119883 119869119880) minus 119892 (A119883119869119880A

119869119883119880)

+ 119892 (A119883119880A119869119883119869119880) minus 119892 ((nabla

119869119880A)119883

119869119883119880)

+ 119892 (T119869119880119883T119880(119869119883))

minus 119892 (T119880119883T119869119880(119869119883))

(96)

119861 (119883 119884) = 119861ℎ

(119883 119884) minus 2119892 (A119883(119869119883) A

119884(119869 119884))

+ 119892 (A119869119883119884A119883(119869 119884))

minus 119892 (A119883119884A119869119883(119869 119884))

(97)

Proof Using Definitions (93) and (94) of paraholomorphicsectional curvature and fundamental equations of submer-sion obtained by OrsquoNeill [1] we have (95) (96) and (97)

Corollary 26 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion from an almostpara-Hermitian manifold onto an almost paracontact pseudo-metric manifold If the fibres of 119891 are totally geodesic pseudo-Riemannian submanifolds of119872 then for any non-lightlike unitvertical vector fields 119880 and 119881 one has

119861 (119880119881) = 119861V(119880 119881) (98)

Corollary 27 Let 119891 119872 rarr 119872 be a paracomplex para-contact pseudo-Riemannian submersion from an almost para-Hermitian manifold onto an almost paracontact pseudometricmanifold and let the fibres of 119891 be totally geodesic pseudo-Riemannian submanifolds of119872 If the horizontal distributionis integrable then for any non-lightlike unit horizontal vectorfields119883 and 119884 one has

119861 (119883 119884) = 119861ℎ

(119883 119884) (99)

Proposition 28 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion from an almostpara-Hermitian manifold onto an almost paracontact pseudo-metric manifold and let the fibres of 119891 be pseudo-Riemanniansubmanifolds of119872 Let 119880 and 119883 be non-lightlike unit verticalvector field and non-lightlike unit horizontal vector fieldrespectively Then one has

119867(119880) = 119867V(119880) +

10038171003817100381710038171003817T119880(119869119880)

10038171003817100381710038171003817

2

minus 119892 (T119869119880(119869119880) T

119880119880)

(100)

119867(119883) = 119867 (119883) ∘ 119891 minus 310038171003817100381710038171003817A119883(119869119883)

10038171003817100381710038171003817

2

(101)

Proof The proof is straightforward If we take 119880 = 119881 in (95)and119883 = 119884 in (97) we have (98) and (99)

Corollary 29 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion from an almostpara-Hermitian manifold onto an almost paracontact pseudo-metric manifold If the fibres of 119891 are totally geodesic pseudo-Riemannian submanifolds of 119872 then the total manifold andfibres of119891have the same paraholomorphic sectional curvatures

Proof Since the fibres are totally geodesic T = 0 conse-quently we have

119867(119880) = 119867V(119880) (102)

Corollary 30 Let 119891 119872 rarr 119872 be a paracomplex para-contact pseudo-Riemannian submersion from an almost para-Hermitian manifold onto an almost paracontact pseudometricmanifold and let the fibres of 119891 be totally geodesic pseudo-Riemannian submanifolds of119872 If the horizontal distributionis integrable then the base manifold and horizontal distribu-tion have the same paraholomorphic sectional curvatures

Proof Since the horizontal distribution is integrableA = 0consequently we have

119867(119883) = 119867 (119883) ∘ 119891 (103)

Theorem 31 Let 119891 119872119898

rarr 119872119899 be a paracomplex

paracontact pseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometricmanifold 119872 and let the fibres of 119891 be pseudo-Riemanniansubmanifolds of 119872 If 119880 119881 are the non-lightlike unit verticalvector fields and 119883 119884 are the non-lightlike unit horizontalvector fields then one has

119861 (119880119881) = 119861V(119880 119881) (104)

119861 (119883119880) = minus210038171003817100381710038171003817T11988011988310038171003817100381710038171003817

2

(105)

119861 (119883 119884) = 119861 (119883 119884) ∘ 119891 (106)

Proof Using results of Lemma 19 in (95) we have

119861 (119880119881) = 119861V(119880 119881) minus 119892 (119869 (T

119880119881) 119869 (T

119880119881))

minus 119892 (1198692

(T119880119881) T

119880119881)

= 119861 (119880119881) + 119892 (T119880119881T119880119881) minus 119892 (T

119880119881T119880119881)

= 119861V(119880 119881)

(107)

Applying results of Lemma 19 in (96) we have

119861 (119883119880) = 119892 ((nabla119880A)119883

(119869119883) 119869119880) minus 119892 ((nabla119869119880A)119883

(119869119883) 119880)

+ 21003817100381710038171003817A119883119880

1003817100381710038171003817

2

minus 210038171003817100381710038171003817T11988011988310038171003817100381710038171003817

2

(108)

Since byTheorem 21 the horizontal distribution is integrablewe haveA = 0 which implies

119861 (119883119880) = minus210038171003817100381710038171003817T11988011988310038171003817100381710038171003817

2

(109)

In view ofA = 0 (104) follows from (97)

12 Geometry

Theorem 32 Let 119891 119872119898

rarr 119872119899 be a paracomplex

paracontact pseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometricmanifold 119872 and let the fibres of 119891 be pseudo-Riemanniansubmanifolds of119872 If 119880 119883 are non-lightlike unit vertical andnon-lightlike unit horizontal vector fields respectively then onehas

119867(119880) = 119867V(119880) minus 2

1003817100381710038171003817T1198801198801003817100381710038171003817

2

(110)

119867(119883) = 119867 (119883) ∘ 119891 (111)

Proof Since 119891 is the paracomplex paracontact pseudo-Riemannian submersion from a para-Kahler manifold 119872

onto an almost paracontact pseudometric manifold 119872 by(16) and equations of Lemma 19 we have

119892 (T119869119880(119869119880) T

119880119880) = 119892 (119869

2

(T119880119880) T

119880119880) =

1003817100381710038171003817T1198801198801003817100381710038171003817

2

119892 (T119880(119869119880) T

119880(119869119880)) = minus119892 (T

119880119880T119880119880) = minus

1003817100381710038171003817T1198801198801003817100381710038171003817

2

(112)

and by using the above results in (100) we have

119867(119880) = 119867V(119880) minus

1003817100381710038171003817T1198801198801003817100381710038171003817

2

minus1003817100381710038171003817T119880119880

1003817100381710038171003817

2

= 119867V(119880) minus 2

1003817100381710038171003817T1198801198801003817100381710038171003817

2

(113)

Again since horizontal distribution is integrable we haveA = 0 and putting it in (101) we obtain (111)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Uma Shankar Verma is thankful to University Grant Com-mission New Delhi India for financial support

References

[1] B OrsquoNeill ldquoThe fundamental equations of a submersionrdquo TheMichigan Mathematical Journal vol 13 pp 459ndash469 1966

[2] B OrsquoNeill Semi-Riemannian Geometry with Applications toRelativity vol 103 of Pure and Applied Mathematics AcademicPress New York NY USA 1983

[3] A Gray ldquoPseudo-Riemannian almost product manifolds andsubmersionsrdquo vol 16 pp 715ndash737 1967

[4] J P Bourguignon and H B Lawson A Mathematicians Visitto Kaluza-Klein Theory Rendiconti del Seminario Matematico1989

[5] S Ianus and M Visinescu ldquoSpace-time compactification andRiemannian submersionsrdquo in The Mathematical Heritage GRassias and C F Gauss Eds pp 358ndash371 World ScientificRiver Edge NJ USA 1991

[6] J P Bourguignon and H B Lawson Jr ldquoStability and isolationphenomena for Yang-Mills fieldsrdquo Communications in Mathe-matical Physics vol 79 no 2 pp 189ndash230 1981

[7] B Watson ldquoG 1198661015840

-Riemannian submersions and nonlineargauge field equations of general relativityrdquo in Global AnalysismdashAnalysis on Manifolds T Rassias and M Morse Eds vol 57 ofTeubner-Texte zur Mathematik pp 324ndash349 Teubner LeipzigGermany 1983

[8] C Altafini ldquoRedundant robotic chains on Riemannian submer-sionsrdquo IEEE Transactions on Robotics and Automation vol 20no 2 pp 335ndash340 2004

[9] M T Mustafa ldquoApplications of harmonic morphisms to grav-ityrdquo Journal of Mathematical Physics vol 41 no 10 pp 6918ndash6929 2000

[10] B Watson ldquoAlmost Hermitian submersionsrdquo Journal of Differ-ential Geometry vol 11 no 1 pp 147ndash165 1976

[11] D Chinea ldquoAlmost contact metric submersionsrdquo Rendiconti delCircolo Matematico di Palermo II vol 34 no 1 pp 319ndash3301984

[12] D Chinea ldquoTransference of structures on almost complexcontact metric submersionsrdquo Houston Journal of Mathematicsvol 14 no 1 pp 9ndash22 1988

[13] Y Gunduzalp and B Sahin ldquoPara-contact para-complexpseudo-Riemannian submersionsrdquo Bulletin of the MalaysianMathematical Sciences Society

[14] I Sato ldquoOn a structure similar to the almost contact structurerdquoTensor vol 30 no 3 pp 219ndash224 1976

[15] P K Rashevskij ldquoThe scalar field in a stratified spacerdquoTrudy Seminara po Vektornomu i Tenzornomu Analizu s ikhPrilozheniyami k Geometrii Mekhanike i Fizike vol 6 pp 225ndash248 1948

[16] P Libermann ldquoSur les structures presque paracomplexesrdquoComptes Rendus de lrsquoAcademie des Sciences I vol 234 pp 2517ndash2519 1952

[17] E M Patterson ldquoRiemann extensions which have Kahlermetricsrdquo Proceedings of the Royal Society of Edinburgh AMathematics vol 64 pp 113ndash126 1954

[18] S Sasaki ldquoOn differentiable manifolds with certain structureswhich are closely related to almost contact structure Irdquo TheTohoku Mathematical Journal vol 12 pp 459ndash476 1960

[19] S Zamkovoy ldquoCanonical connections on paracontact mani-foldsrdquo Annals of Global Analysis and Geometry vol 36 no 1pp 37ndash60 2009

[20] D E Blair Riemannian Geometry of Contact and SymplecticManifolds vol 23 of Progress in Mathematics BirkhauserBoston Mass USA 2002

[21] V Cruceanu P Fortuny and P M Gadea ldquoA survey onparacomplex geometryrdquoThe Rocky Mountain Journal of Mathe-matics vol 26 no 1 pp 83ndash115 1996

[22] P M Gadea and J M Masque ldquoClassification of almost para-Hermitian manifoldsrdquo Rendiconti di Matematica e delle sueApplicazioni VII vol 7 no 11 pp 377ndash396 1991

[23] M Falcitelli S Ianus and A M Pastore Riemannian Submer-sions and Related Topics World Scientific River Edge NJ USA2004

[24] E G Rio and D N Kupeli Semi-Riemannian Maps andTheir Applications Kluwer Academic Publisher DordrechtTheNetherlands 1999

[25] B Sahin ldquoSemi-invariant submersions from almost HermitianmanifoldsrdquoCanadianMathematical Bulletin vol 54 no 3 2011

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 6: Research Article Paracomplex Paracontact Pseudo-Riemannian …downloads.hindawi.com/archive/2014/616487.pdf · 2019-07-31 · Research Article Paracomplex Paracontact Pseudo-Riemannian

6 Geometry

paracontact pseudo-Riemannian manifold Consider a sub-mersion 119891 R6

3 (1199091 1199092 1199093 1199101 1199102 1199103)119905

rarr R31 (119906 V 119908)119905

defined by

119891 ((1199091 1199092 1199093 1199101 1199102 1199103)119905

)

997891997888rarr (1199091+ 1199092

radic2

1199101+ 1199102

radic2

1199102+ 1199103

radic2

)

119905

(57)

Then there exists one form 120578 = (1198891199092+ 1198891199093)radic2 on (R6

3 119869 119892)

such that (39) is satisfied The kernel of 119891lowastis

V = ker119891lowast

= Span1198811=

120597

1205971199091

minus120597

1205971199092

1198812=

120597

1205971199101

minus120597

1205971199102

+120597

1205971199103

1198813=

120597

1205971199093

(58)

which is vertical distribution admitting non-lightlike vectorfields that is the fibre is nondegenerate submanifold of(R63 119869 119892)The horizontal distribution is

H = Span1198831=

120597

1205971199091

+120597

1205971199092

1198832= minus

120597

1205971199101

+120597

1205971199103

1198833=

120597

1205971199101

+120597

1205971199102

(59)

Example 10 Let (R42 119869 119892) be a paracomplex pseudometric

manifold and let (R31 120601 120585 120578 119892) be an almost paracontact

pseudometric manifoldConsider a submersion 119891 R4

2 (1199091 1199092 1199101 1199102)119905

rarr R31

(119906 V 119908)119905 defined by

119891 ((1199091 1199092 1199101 1199102)119905

) 997891997888rarr (1199091 1199101 1199102)119905

(60)

Then the kernel of 119891lowastis

V = ker119891lowast= Span119881

1=

120597

1205971199092

(61)

which is the vertical distribution and the restriction of 119892 tothe fibres of 119891 is nondegenerate

The horizontal distribution is

H = (ker119891lowast)perp

= Span119883 =120597

1205971199091

119884 =120597

1205971199101

120585ℎ

=120597

1205971199102

(62)

The characteristic vector field 120585 = 120597120597119908 on R31has unique

horizontal lift 120585ℎ

which is the characteristic vector field onhorizontal distributionH of R4

2

We also have

119892 (119883119883) = 119892 (119891lowast119883119891lowast119883) = minus1

119892 (119884 119884) = 119892 (119891lowast119884 119891lowast119884) = 1

119892 (120585ℎ

120585ℎ

) = 119892(119891lowast120585ℎ

119891lowast120585ℎ

) = 119892 (120585 120585) = 1

(63)

Thus the smooth map 119891 is a pseudo-Riemannian submer-sion

Moreover we obtain that there exists a 1-form 120578 = 1198891199092on

R42such that 120578(119869120585

ℎ

) = 1 120578(120585ℎ

) = 0 and the map 119891 satisfies

119891lowast119869119883 = 120601119891

lowast119883 + 120578 (119883) 120585

119891lowast119869119884 = 120601119891

lowast119884 + 120578 (119884) 120585

119891lowast119869120585ℎ

= 120601119891lowast120585ℎ

+ 120578 (120585ℎ

) 120585

(64)

Hence the map 119891 is a paracomplex paracontact pseudo-Ri-emannian submersion from R4

2on to R3

1

Proposition 11 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion and let the fibresof 119891 be pseudo-Riemannian submanifolds of 119872 Let 119883 119884 bebasic vector fields 119891-related to119883 119884 respectively Then 119869(119883) minus120576119892(119883 120585

ℎ

)120585ℎ

is 119891-related to 120601119883

Proof Since119883 is 119891-related to vector field119883 on119872 we have

120578 (119883) = 120578V+ 120578ℎ

(119883) = 0 + 120578ℎ

(119883) = 120576119892 (119883 120585ℎ

)

997904rArr 119891lowast(119869119883) = 120601119883 + 120578

ℎ

(119883) 120585

997904rArr 119891lowast119869119883 minus 120576119892 (119883 120585

ℎ

) 120585ℎ

= 120601119883

(65)

Hence 119869(119883) minus 120576119892(119883 120585ℎ

)120585ℎ

is 119891-related to 120601119883

Proposition 12 Let 119891 119872 rarr 119872 be a paracomplex para-contact pseudo-Riemannian submersion and let the fibres of 119891be pseudo-Riemannian submanifolds of 119872 Let V and H bethe vertical and horizontal distributions respectively If 120585

ℎ

isthe basic characteristic vector field of horizontal distribution119891-related to the characteristic vector field 120585 of base manifoldthen

(i) 119869V sub DVoplus 119869120585ℎ

oplus 120585ℎ

(ii) 119869H sub Dℎ

oplus 120585ℎ

oplus 119869120585ℎ

Proof (i) Let 119880 isin V Then 119880 = 119886119880|DV + 119887119869120585

ℎ

for 119886 119887 isin

119862infin

(119872) as 119869120585ℎ

= 120585Vis characteristic vector field on odd

Geometry 7

dimensional fibre submanifold 119891minus1(119902) of119872 119902 isin 119872 We get

119869119880 = 119886119869119880|DV + 119887119869

2

120585ℎ

= 119886119869119880|DV + 119887120585

ℎ

isinV oplus 120585ℎ

997904rArr 119869V subV oplus 120585ℎ

(66)

Again let 119881 isinV oplus 120585ℎ

Then 119881 = 119886119881|DV + 119887119869120585

ℎ

+ 119888120585ℎ

where

120578V(119881|DV ) = 0 D

V= ker 120578V 119886119881

|DV + 119887119869120585

ℎ

isin V and 119886 119887 119888 isin119862infin

(119872) We have

119869119881 = 119886119869119881|DV + 119887120585

ℎ

+ 119888119869120585ℎ

= (119886119869119881|DV + 119888119869120585

ℎ

)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

isinV

+ 119887120585ℎ

⏟⏟⏟⏟⏟⏟⏟

isin120585

ℎ

isinV oplus 120585ℎ

(67)

Now by (39) we get

119891lowast119869119881 = 120601 (119891

lowast119881) + 120578 (119891

lowast(119881)) 120585

= 119888 120601 (119891lowast120585ℎ

) + 120578 (120585) 120585

= 119888120585 isin 120585 sube 119869V

(68)

We get 119869119881 notinVHence 119869V subV oplus 120585

ℎ

that is 119869V sub DVoplus 119869120585ℎ

oplus 120585ℎ

(ii) Let 119883 = 119886119883

|

Dℎ+ 119887120585ℎ

isin H where H = Dℎ

oplus

120585ℎ

ker 120578ℎ = Dℎ

and 119886 119887 isin 119862infin(119872) Then

119869119883 = 119886119869119883|

Dℎ+ 119887119869120585ℎ

isinH oplus 119869120585ℎ

(69)

which implies that 119869H subH oplus 119869120585ℎ

Again let119884 isinHoplus119869120585

ℎ

Then119884 = 119886119884|

Dℎ+119887120585ℎ

+119888119869120585ℎ

notinHfor 119886 119887 119888 isin 119862infin(119872) We have

119869 119884 = 119886119869119884|

Dℎ+ 119887119869120585ℎ

+ 1198881198692

120585ℎ

= 119886119869119884|

Dℎ+ 119887119869120585ℎ

+ 119888120585ℎ

= 119885 + 119887119869120585ℎ

isinH oplus 119869120585ℎ

for some 119885 = 119886119869119884|

Dℎ+ 119888120585ℎ

isinH

(70)

We obtain 119869 119884 notinHHence 119869H sub H oplus 119869(120585

ℎ

) that is 119869H sub Dℎ

oplus 120585ℎ

oplus

119869120585ℎ

Example 13 Let (R63 119869 119892) be an almost paracomplex pseudo-

Riemannian manifold and let (R31 120601 120585 120578 119892) be an almost

paracontact pseudo-Riemannian manifold Consider a sub-mersion 119891 R6

3 (1199091 1199092 1199093 1199101 1199102 1199103)119905

rarr R31 (119906 V 119908)119905

defined by

119891 ((1199091 1199092 1199093 1199101 1199102 1199103)119905

) 997891997888rarr (1199091+ 1199092

radic2

1199101+ 1199102

radic2

1199103)

119905

(71)

Then the kernel of 119891lowastis

V = ker119891lowast

= Span1198811=

120597

1205971199091

minus120597

1205971199092

1198812=

120597

1205971199101

minus120597

1205971199102

120585V=

120597

1205971199093

(72)

which is the vertical distribution and the restriction of 119892 tothe fibres of 119891 is nondegenerate

The horizontal distribution is

H = (ker119891lowast)perp

= Span1198831=

120597

1205971199091

+120597

1205971199092

1198832=

120597

1205971199101

+120597

1205971199102

120585ℎ

=120597

1205971199103

(73)

The characteristic vector field 120585 = 120597120597119908 on R31has unique

horizontal lift 120585ℎ

which is the characteristic vector field onthe horizontal distributionH of R6

3

We also have

119892 (1198831 1198831) = 119892 (119891

lowast1198831 119891lowast1198831) = minus2

119892 (1198832 1198832) = 119892 (119891

lowast1198832 119891lowast1198832) = 2

119892 (120585ℎ

120585ℎ

) = 119892(119891lowast120585ℎ

119891lowast120585ℎ

) = 119892 (120585 120585) = 1

(74)

Thus the smooth map 119891 is a pseudo-Riemannian submer-sion

Also we obtain that there exists a 1-form 120578 = 1198891199093on R63

such that 120578(119869120585ℎ

) = 1 120578(120585ℎ

) = 0 and the map 119891 satisfies

119891lowast1198691198831= 120601119891lowast1198831+ 120578 (119883

1) 120585

119891lowast1198691198832= 120601119891lowast1198832+ 120578 (119883

2) 120585

119891lowast119869120585ℎ

= 120601119891lowast120585ℎ

+ 120578 (120585ℎ

) 120585

(75)

Hence the map 119891 is a paracomplex paracontact pseudo-Riemannian submersion from R6

3onto R3

1

Moreover we observe that for this submersion119891 we have

119869V subV oplus 120585ℎ

119869H subH oplus 119869120585ℎ

(76)

which verifies Proposition 12

8 Geometry

Proposition 14 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion and let the fibresof 119891 be pseudo-Riemannian submanifolds of 119872 Let 119883 119884 bebasic vector fields 119891-related to 119883 119884 respectively Let 119865 and Φbe the second fundamental forms and let nabla and nabla be the Levi-Civita connection on the total manifold119872 and base manifold119872 respectively Then one has

(i) 119891lowast((nabla119883119869)119884) = (nabla

119883120601)119884 + 120576119892(119884 nabla

119883120585)120585 + 120578(119884)nabla

119883120585

(ii) 119865 = 119891lowastlowastΦ + 120576120578 otimes 120578

(iii) 119891lowast((nabla119883119865)(119884 119885)) = (nabla

119883Φ)(119884 119885) + 120578(119884)119892(119885 nabla

119883120585) +

120578(119885)119892(119884 nabla119883120585)

Proof (i) In view of Definition 2 and Proposition 11 we have

119891lowast((nabla119883119869) 119884) = 119891

lowast(nabla119883(119869 119884) minus 119869 (nabla

119883119884))

= nabla119883(119891lowast(119869 119884)) minus 119891

lowast(119869 (nabla119883119884))

= nabla119883(120601119884) + nabla

119883(120578 (119884) 120585) minus 120601 (nabla

119883119884)

minus 120578 (nabla119883119884) 120585

= (nabla119883120601)119884 + nabla

119883(120576119892 (119884 120585) 120585) minus 120578 (nabla

119883119884) 120585

= (nabla119883120601)119884 + 120576119892 (nabla

119883119884 120585) 120585 + 120576119892 (119884 nabla

119883120585) 120585

+ 120576119892 (119884 120585) nabla119883120585 minus 120576119892 (nabla

119883119884 120585) 120585

= (nabla119883120601)119884 + 120576119892 (119884 nabla

119883120585) 120585 + 120578 (119884) nabla

119883120585

(77)

(ii) Since119891lowastlowastΦ is pullback ofΦ through the linearmap119891

lowast

we get

119891lowast

lowastΦ(119883 119884) = Φ (119883 119884) ∘ 119891 = 119892 (119883 120601119884) ∘ 119891

= 119892 (119883 119869 119884) minus 120576120578 (119884) 120578 (119883)

= 119865 (119883 119884) minus 120576120578 (119883) 120578 (119884)

(78)

which implies 119865 = 119891lowastlowastΦ + 120576120578 otimes 120578

(iii) By (23) we have

119891lowast((nabla119883119865) (119884 119885)) = 119892 (119891

lowast(119884) 119891

lowast((nabla119883119869)119885)) (79)

Now using (i) in the above equation we get (iii)

Theorem 15 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion and let the fibresof 119891 be pseudo-Riemannian submanifolds of 119872 Let 119883 119884 bebasic vector fields 119891-related to 119883 119884 respectively If the totalspace is para-Hermitian manifold then the almost paracontactstructure of base space is normal

Moreover if the almost paracontact structure of base spaceis normal then the Nijenhuis tensor of total space is vertical

Proof The Nijenhuis tensors 119873119869and 119873

120601of almost para-

complex structure 119869 and almost paracontact structure 120601 arerespectively defined by (8) and (11)

Using Definition 2 and properties of Sections 21 and 22we get the following identity

119891lowast(119873119869(119883 119884)) = 119873

(1)

(119883 119884) + 2119889120578 (120601119883 119884) 120585

minus 2119889120578 (120601119884119883) 120585

+ 2120578 (119883) 119889120578 (120585 119884) 120585 minus 2120578 (119884) 119889120578 (120585 119883) 120585

minus 120578 (119884)119873(3)

(119883) + 120578 (119883)119873(3)

(119884)

(80)

Using (12) (13) (14) and (15) (80) reduces to

119891lowast(119873119869(119883 119884)) = 119873

(1)

(119883 119884) + 119873(2)

(119883 119884) 120585

+ 120578 (119883)119873(4)

(119884) 120585

minus 120578 (119884)119873(4)

(119883) 120585 minus 120578 (119884)119873(3)

(119883)

+ 120578 (119883)119873(3)

(119884)

(81)

Since 119873119869(119883 119884) = 0 it follows from (81) that

tensors 119873(1) 119873(2) 119873(3) and 119873(4) vanish togetherHence the almost paracontact structure of base space is

normalConversely let the almost paracontact structure of the

base space be normalThen (81) implies that 119891

lowast(119873119869(119883 119884)) = 0

Hence119873119869(119883 119884) is vertical

Corollary 16 Let119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion and let the fibres of 119891 bepseudo-Riemannian submanifolds of 119872 Let 119883 119884 be basicvector fields 119891-related to 119883 119884 respectively Let the total spacebe para-Hermitian manifold and119873(1) vanishes Then the basespace is paracontact pseudometric manifold if and only if 120585 iskilling

Proof Let the total space be para-Hermitian and 119873(1) van-ishes Then from (80) we have

0 = 2119889120578 (120601119883 119884) 120585 minus 2119889120578 (120601119884119883) 120585 + 2120578 (119883) 119889120578 (120585 119884) 120585

minus 2120578 (119884) 119889120578 (120585 119883) 120585 minus 120578 (119884) (L120585120601)119883 + 120578 (119883) (L

120585120601)119884

(82)

If 120585 is killing then we have L120585120601 = 0 It immediately follows

from (82) that

119889120578 (120601119883 119884) minus 120578 (120601119884119883) + 120578 (119883) 119889120578 (120585 119884)

minus 120578 (119884) 119889120578 (120585 119883) = 0

(83)

In view of (6) and (7) the above equation gives 119889120578 = ΦConversely let the base space be paracontact Then 119889120578 =

ΦUsing (6) (7) and (82) we getL

120585120601 = 0

Hence the characteristic vector field 120585 is killing

Geometry 9

Theorem 17 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion and let the fibres of 119891 bepseudo-Riemannian submanifolds of 119872 Let 119883 119884 be basicvector fields 119891-related to 119883 119884 respectively If the total spaceis para-Kahler then the base space is paracosymplectic Theconverse is true if nabla

119883119869 is vertical

Proof We have for any 119883119884 isin Γ(119879119872) (nabla119883120601)119884 = 0 which

gives 119892(119885 (nabla119883120601)119884) = 0 for any 119885 isin Γ(119879119872)

From Proposition 14 we have

119892 (119885 (nabla119883120601)119884) + 120576120578 (119884) (nabla

119883120578)119885 + 120576120578 (119885) (nabla

119883120578) 119884 ∘ 119891

= 119892 (119885 (nabla119883119869) 119884)

(84)

Let nabla 119869 = 0 that is the total space is para-KahlerThen from(84) we obtain nabla120601 = 0 and nabla120578 = 0 Hence the base space isparacosymplectic

Again let (nabla119883120601)119884 = 0 and nabla120578 = 0 Then 119892(119885 (nabla

119883119869)119884) =

0 which implies that (nabla119883119869)119884 is a vertical vector field

Theorem 18 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion and let the fibres of 119891 bepseudo-Riemannian submanifolds of 119872 Let 119883 119884 and 119885 bebasic vector fields 119891-related to 119883 119884 and 119885 respectively If120578(119883)119885(120578(119884))+120578(119884)119883(120578(119885))minus120578(119883)119884(120578(119885)) = 0 then the totalspace is almost para-Kahler if and only if the base space119872 isan almost paracosymplectic manifold

Proof We have the following equation

3119889119865 (119883 119884 119885)

= 3 (119891lowast

lowast119889Φ) (119883 119884 119885) + 2120576120578 (119885) 119889120578 (119883 119884)

minus 2120576120578 (119884) 119889120578 (119883 119885)

+ 2120576120578 (119883) 119889120578 (119884 119885) + 2120576120578 (119883)119885 (120578 (119884))

+ 2120576120578 (119884)119883 (120578 (119885))

minus 2120576120578 (119883)119884 (120578 (119885))

(85)

If 119889120578 = 0 119889Φ = 0 and 120578(119883)119885(120578(119884)) + 120578(119884)119883(120578(119885)) minus

120578(119883)119884(120578(119885)) = 0 then from (85) we have 119889119865 = 0 Hencethe total space is almost para-Kahler

Conversely let 119889119865 = 0 and 120578(119883)119885(120578(119884))+120578(119884)119883(120578(119885))minus120578(119883)119884(120578(119885)) = 0

By using the above equation in (85) we have 119889120578 = 0 and119889Φ = 0

Hence the base space is almost paracosymplectic

Nowwe investigate the properties of fundamental tensorsT andA of a pseudo-Riemannian submersion

Lemma 19 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometric manifold119872 and

let the fibres of 119891 be pseudo-Riemannian submanifolds of 119872Then for any horizontal vector fields119883 119884 and for any verticalvector fields 119880 119881 on119872 one has

(i) A119883(119869 119884) = 119869(A

119883119884)

(ii) A119869119883(119884) = 119869(A

119883119884)

(iii) T119880(119869 119881) = 119869(T

119880119881)

(iv) T119869119880119881 = 119869(T

119880119881)

Proof The proof follows using similar steps as in Lemmas 3and 4 of [13] so we omit it

Lemma 20 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometric manifold119872 andlet the fibres of 119891 be pseudo-Riemannian submanifolds of 119872Then for any vector fields 119864 119865 on119872 one has

(i) A119864(119869119865) = 119869(A

119864119865)

(ii) T119864(119869119865) = 119869(T

119864119865)

Proof The proof follows from (37) and (38)

Theorem 21 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometric manifold119872 andlet the fibres of 119891 be pseudo-Riemannian submanifolds of 119872Then the horizontal distribution is integrable

Proof For any vertical vector field 119880 we have

119892 (119869 (A119883119884) 119880) = 119892 (A

119883119869 119884119880)

= minus119892 (119869119884A119883119880)

= minus119892 (119869119884 ℎ (nabla119880119883))

= 119892 (119884 ℎ (119869 (nabla119880119883)))

= 119892 (119884 ℎ (minusnabla119880119869)119883

+nabla119880(119869119883))

= 119892 (119884 ℎ nabla119880(119869119883)) = 119892 (119884A

119869119883119880)

= minus119892 (A119869119883119884119880) = minus119892 (119869 (A

119883119884) 119880)

(86)

Thus 119892(119869(A119883119884) 119880) = 0 which is true for all119883 and 119884

SoA119883119884 = 0

Hence the horizontal distribution is integrable

Theorem 22 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometric manifold119872 andlet the fibres of 119891 be pseudo-Riemannian submanifolds of 119872Then the submersion is an affine map onH

10 Geometry

Proof The second fundamental form of 119891 is defined by

(nabla119891lowast) (119864 119865) = (nabla

119891

119864119891lowast(119865)) ∘ 119891 minus 119891

lowast(nabla119864119865) (87)

where 119864 119865 isin Γ(119879119872) and nabla119891 is pullback connection of Levi-Civita connection nabla on119872 with respect to 119891

We have for any119883119884 isinH

(nabla119891lowast) (119883 119884) = (nabla

119891

119883

119891lowast(119884)) ∘ 119891 minus 119891

lowast(nabla119883119884) (88)

By using Lemma 1 we have 119891lowast(ℎ(nabla119883119884)) = (nabla

119883119884) ∘ 119891 which

implies nabla119891lowast= 0

Hence the submersion 119891 is an affine map onH

Theorem 23 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Hermitian man-ifold119872 onto an almost paracontact pseudometric manifold119872and let the fibres of 119891 be pseudo-Riemannian submanifolds of119872 Then the submersion is an affine map on V if and only ifthe fibres of 119891 are totally geodesic

Proof We have for any 119880119881 isinV

(nabla119891lowast) (119880 119881) = minus119891

lowast(ℎ (nabla119880119881)) (89)

which in view of (27) gives

(nabla119891lowast) (119880 119881) = minus119891

lowast(T119880119881) (90)

Let the fibres of 119891 be totally geodesic Then T = 0Consequently from the above equation we have nabla119891

lowast= 0

Thus the map 119891 is affine onVConversely let the submersion 119891 be an affine map onV

Then nabla119891lowast= 0 which impliesT = 0

Hence the fibres of 119891 are totally geodesic

Theorem 24 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Hermitian man-ifold119872 onto an almost paracontact pseudometric manifold119872and let the fibres of 119891 be pseudo-Riemannian submanifoldsof 119872 Then the submersion is an affine map if and only ifℎ(nabla119864ℎ119865)+A

ℎ119864V119865+TV119864V119865 is 119891-related to nabla

119883119884 for any 119864 119865 isin

Γ(119879119872)

Proof For any 119864 119865 isin Γ(119879119872) with 119891lowastℎ119864 = 119883 ∘ 119891 and 119891

lowastV119865 =

119884 ∘ 119891 we have

(nabla119891lowast) (119864 119865) = (nabla

119891lowastℎ119864(119891lowastℎ119865)) ∘ 119891 minus 119891

lowast(ℎ (nabla119864119865))

= (nabla119883119884) ∘ 119891 minus 119891

lowast(ℎ (nablaℎ119864ℎ119865 + nabla

ℎ119864V119865

+nablaV119864ℎ119865 + nablaV119864ℎ119865))

(91)

By using (27) and (31) in the above equation we have

(nabla119891lowast) (119864 119865) = (nabla

119883119884) ∘ 119891 minus 119891

lowast(ℎ (nabla119864ℎ119865) +A

ℎ119864V119865

+TV119864V119865) (92)

Let the submersion map be affine Then for any 119864 119865 isin

Γ(119879119872) (nabla119891lowast)(119864 119865) = 0 Equation (92) implies (nabla

119883119884) ∘ 119891 =

119891lowast(ℎ(nabla119864ℎ119865) +A

ℎ119864V119865 +TV119864ℎ119865)

Conversely let ℎ(nabla119864119865) + A

ℎ119864V119865 + TV119864ℎ119865 be 119891-related

to nabla119883119884 for any 119864 119865 isin Γ(119879119872) Then from (92) we have

(nabla119891lowast)(119864 119865) = 0Hence the submersion map 119891 is affine

4 Curvature Properties

In this section the paraholomorphic bisectional curvaturesand paraholomorphic sectional curvatures of total mani-fold base manifold and fibres of paracomplex paracontactpseudo-Riemannian submersion and their curvature proper-ties are studied

Let 119891 119872 rarr 119872 be a paracomplex paracontact pseudo-Riemannian submersion from an almost para-Hermitianmanifold (119872 119869 119892) onto an almost paracontact pseudometricmanifold (119872 120601 120585 120578 119892)

Suppose that the vector fields 119864 119865 span the 2-dimensional plane at point 119901 of 119872 and let R be theRiemannian curvature tensor of 119872 The paraholomorphicbisectional curvature 119861(119864 119865) of 119872 for any pair of nonzeronon-lightlike vector fields 119864 119865 on 119872 is defined by theformula

119861 (119864 119865) =

R (119864 119869119864 119865 119869119865)

119892 (119864 119864) 119892 (119865 119865) (93)

For a nonzero non-lightlike vector field 119864 the vector field119869119864 is also non-lightlike and 119864 119869119864 span the 2-dimensionalplane Then the paraholomorphic sectional curvature 119867(119864)is defined as

119867(119864) = 119861 (119864 119864) =

R (119864 119869119864 119864 119869119864)

119892 (119864 119864) 119892 (119864 119864) (94)

The curvature properties of Riemannian submersion andsemi-Riemannian submersion have been extensively studiedin the work of OrsquoNeill [1] and Gray [3]

Let 119861ℎ and 119861V be the paraholomorphic bisectional cur-vatures of horizontal and vertical spaces respectively Let119867ℎ and 119867V be the paraholomorphic sectional curvatures of

horizontal and vertical spaces respectively Let 119861 and 119867 bethe paraholomorphic bisectional and sectional curvatures ofthe base manifold respectively

Proposition 25 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion from an almostpara-Hermitian manifold onto an almost paracontact pseudo-metric manifold and let the fibres of 119891 be pseudo-Riemanniansubmanifolds of 119872 Let 119880 119881 be non-lightlike unit verticalvector fields and let119883 119884 be non-lightlike unit horizontal vectorfields on119872 Then one has

119861 (119880119881) = 119861V(119880 119881) + 119892 (T

119880(119869119881) T

119869119880119881)

minus 119892 (T119869119880(119869119881) T

119880119881)

(95)

Geometry 11

119861 (119883119880) = 119892 ((nabla119880A)119883

119869119883 119869119880) minus 119892 (A119883119869119880A

119869119883119880)

+ 119892 (A119883119880A119869119883119869119880) minus 119892 ((nabla

119869119880A)119883

119869119883119880)

+ 119892 (T119869119880119883T119880(119869119883))

minus 119892 (T119880119883T119869119880(119869119883))

(96)

119861 (119883 119884) = 119861ℎ

(119883 119884) minus 2119892 (A119883(119869119883) A

119884(119869 119884))

+ 119892 (A119869119883119884A119883(119869 119884))

minus 119892 (A119883119884A119869119883(119869 119884))

(97)

Proof Using Definitions (93) and (94) of paraholomorphicsectional curvature and fundamental equations of submer-sion obtained by OrsquoNeill [1] we have (95) (96) and (97)

Corollary 26 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion from an almostpara-Hermitian manifold onto an almost paracontact pseudo-metric manifold If the fibres of 119891 are totally geodesic pseudo-Riemannian submanifolds of119872 then for any non-lightlike unitvertical vector fields 119880 and 119881 one has

119861 (119880119881) = 119861V(119880 119881) (98)

Corollary 27 Let 119891 119872 rarr 119872 be a paracomplex para-contact pseudo-Riemannian submersion from an almost para-Hermitian manifold onto an almost paracontact pseudometricmanifold and let the fibres of 119891 be totally geodesic pseudo-Riemannian submanifolds of119872 If the horizontal distributionis integrable then for any non-lightlike unit horizontal vectorfields119883 and 119884 one has

119861 (119883 119884) = 119861ℎ

(119883 119884) (99)

Proposition 28 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion from an almostpara-Hermitian manifold onto an almost paracontact pseudo-metric manifold and let the fibres of 119891 be pseudo-Riemanniansubmanifolds of119872 Let 119880 and 119883 be non-lightlike unit verticalvector field and non-lightlike unit horizontal vector fieldrespectively Then one has

119867(119880) = 119867V(119880) +

10038171003817100381710038171003817T119880(119869119880)

10038171003817100381710038171003817

2

minus 119892 (T119869119880(119869119880) T

119880119880)

(100)

119867(119883) = 119867 (119883) ∘ 119891 minus 310038171003817100381710038171003817A119883(119869119883)

10038171003817100381710038171003817

2

(101)

Proof The proof is straightforward If we take 119880 = 119881 in (95)and119883 = 119884 in (97) we have (98) and (99)

Corollary 29 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion from an almostpara-Hermitian manifold onto an almost paracontact pseudo-metric manifold If the fibres of 119891 are totally geodesic pseudo-Riemannian submanifolds of 119872 then the total manifold andfibres of119891have the same paraholomorphic sectional curvatures

Proof Since the fibres are totally geodesic T = 0 conse-quently we have

119867(119880) = 119867V(119880) (102)

Corollary 30 Let 119891 119872 rarr 119872 be a paracomplex para-contact pseudo-Riemannian submersion from an almost para-Hermitian manifold onto an almost paracontact pseudometricmanifold and let the fibres of 119891 be totally geodesic pseudo-Riemannian submanifolds of119872 If the horizontal distributionis integrable then the base manifold and horizontal distribu-tion have the same paraholomorphic sectional curvatures

Proof Since the horizontal distribution is integrableA = 0consequently we have

119867(119883) = 119867 (119883) ∘ 119891 (103)

Theorem 31 Let 119891 119872119898

rarr 119872119899 be a paracomplex

paracontact pseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometricmanifold 119872 and let the fibres of 119891 be pseudo-Riemanniansubmanifolds of 119872 If 119880 119881 are the non-lightlike unit verticalvector fields and 119883 119884 are the non-lightlike unit horizontalvector fields then one has

119861 (119880119881) = 119861V(119880 119881) (104)

119861 (119883119880) = minus210038171003817100381710038171003817T11988011988310038171003817100381710038171003817

2

(105)

119861 (119883 119884) = 119861 (119883 119884) ∘ 119891 (106)

Proof Using results of Lemma 19 in (95) we have

119861 (119880119881) = 119861V(119880 119881) minus 119892 (119869 (T

119880119881) 119869 (T

119880119881))

minus 119892 (1198692

(T119880119881) T

119880119881)

= 119861 (119880119881) + 119892 (T119880119881T119880119881) minus 119892 (T

119880119881T119880119881)

= 119861V(119880 119881)

(107)

Applying results of Lemma 19 in (96) we have

119861 (119883119880) = 119892 ((nabla119880A)119883

(119869119883) 119869119880) minus 119892 ((nabla119869119880A)119883

(119869119883) 119880)

+ 21003817100381710038171003817A119883119880

1003817100381710038171003817

2

minus 210038171003817100381710038171003817T11988011988310038171003817100381710038171003817

2

(108)

Since byTheorem 21 the horizontal distribution is integrablewe haveA = 0 which implies

119861 (119883119880) = minus210038171003817100381710038171003817T11988011988310038171003817100381710038171003817

2

(109)

In view ofA = 0 (104) follows from (97)

12 Geometry

Theorem 32 Let 119891 119872119898

rarr 119872119899 be a paracomplex

paracontact pseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometricmanifold 119872 and let the fibres of 119891 be pseudo-Riemanniansubmanifolds of119872 If 119880 119883 are non-lightlike unit vertical andnon-lightlike unit horizontal vector fields respectively then onehas

119867(119880) = 119867V(119880) minus 2

1003817100381710038171003817T1198801198801003817100381710038171003817

2

(110)

119867(119883) = 119867 (119883) ∘ 119891 (111)

Proof Since 119891 is the paracomplex paracontact pseudo-Riemannian submersion from a para-Kahler manifold 119872

onto an almost paracontact pseudometric manifold 119872 by(16) and equations of Lemma 19 we have

119892 (T119869119880(119869119880) T

119880119880) = 119892 (119869

2

(T119880119880) T

119880119880) =

1003817100381710038171003817T1198801198801003817100381710038171003817

2

119892 (T119880(119869119880) T

119880(119869119880)) = minus119892 (T

119880119880T119880119880) = minus

1003817100381710038171003817T1198801198801003817100381710038171003817

2

(112)

and by using the above results in (100) we have

119867(119880) = 119867V(119880) minus

1003817100381710038171003817T1198801198801003817100381710038171003817

2

minus1003817100381710038171003817T119880119880

1003817100381710038171003817

2

= 119867V(119880) minus 2

1003817100381710038171003817T1198801198801003817100381710038171003817

2

(113)

Again since horizontal distribution is integrable we haveA = 0 and putting it in (101) we obtain (111)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Uma Shankar Verma is thankful to University Grant Com-mission New Delhi India for financial support

References

[1] B OrsquoNeill ldquoThe fundamental equations of a submersionrdquo TheMichigan Mathematical Journal vol 13 pp 459ndash469 1966

[2] B OrsquoNeill Semi-Riemannian Geometry with Applications toRelativity vol 103 of Pure and Applied Mathematics AcademicPress New York NY USA 1983

[3] A Gray ldquoPseudo-Riemannian almost product manifolds andsubmersionsrdquo vol 16 pp 715ndash737 1967

[4] J P Bourguignon and H B Lawson A Mathematicians Visitto Kaluza-Klein Theory Rendiconti del Seminario Matematico1989

[5] S Ianus and M Visinescu ldquoSpace-time compactification andRiemannian submersionsrdquo in The Mathematical Heritage GRassias and C F Gauss Eds pp 358ndash371 World ScientificRiver Edge NJ USA 1991

[6] J P Bourguignon and H B Lawson Jr ldquoStability and isolationphenomena for Yang-Mills fieldsrdquo Communications in Mathe-matical Physics vol 79 no 2 pp 189ndash230 1981

[7] B Watson ldquoG 1198661015840

-Riemannian submersions and nonlineargauge field equations of general relativityrdquo in Global AnalysismdashAnalysis on Manifolds T Rassias and M Morse Eds vol 57 ofTeubner-Texte zur Mathematik pp 324ndash349 Teubner LeipzigGermany 1983

[8] C Altafini ldquoRedundant robotic chains on Riemannian submer-sionsrdquo IEEE Transactions on Robotics and Automation vol 20no 2 pp 335ndash340 2004

[9] M T Mustafa ldquoApplications of harmonic morphisms to grav-ityrdquo Journal of Mathematical Physics vol 41 no 10 pp 6918ndash6929 2000

[10] B Watson ldquoAlmost Hermitian submersionsrdquo Journal of Differ-ential Geometry vol 11 no 1 pp 147ndash165 1976

[11] D Chinea ldquoAlmost contact metric submersionsrdquo Rendiconti delCircolo Matematico di Palermo II vol 34 no 1 pp 319ndash3301984

[12] D Chinea ldquoTransference of structures on almost complexcontact metric submersionsrdquo Houston Journal of Mathematicsvol 14 no 1 pp 9ndash22 1988

[13] Y Gunduzalp and B Sahin ldquoPara-contact para-complexpseudo-Riemannian submersionsrdquo Bulletin of the MalaysianMathematical Sciences Society

[14] I Sato ldquoOn a structure similar to the almost contact structurerdquoTensor vol 30 no 3 pp 219ndash224 1976

[15] P K Rashevskij ldquoThe scalar field in a stratified spacerdquoTrudy Seminara po Vektornomu i Tenzornomu Analizu s ikhPrilozheniyami k Geometrii Mekhanike i Fizike vol 6 pp 225ndash248 1948

[16] P Libermann ldquoSur les structures presque paracomplexesrdquoComptes Rendus de lrsquoAcademie des Sciences I vol 234 pp 2517ndash2519 1952

[17] E M Patterson ldquoRiemann extensions which have Kahlermetricsrdquo Proceedings of the Royal Society of Edinburgh AMathematics vol 64 pp 113ndash126 1954

[18] S Sasaki ldquoOn differentiable manifolds with certain structureswhich are closely related to almost contact structure Irdquo TheTohoku Mathematical Journal vol 12 pp 459ndash476 1960

[19] S Zamkovoy ldquoCanonical connections on paracontact mani-foldsrdquo Annals of Global Analysis and Geometry vol 36 no 1pp 37ndash60 2009

[20] D E Blair Riemannian Geometry of Contact and SymplecticManifolds vol 23 of Progress in Mathematics BirkhauserBoston Mass USA 2002

[21] V Cruceanu P Fortuny and P M Gadea ldquoA survey onparacomplex geometryrdquoThe Rocky Mountain Journal of Mathe-matics vol 26 no 1 pp 83ndash115 1996

[22] P M Gadea and J M Masque ldquoClassification of almost para-Hermitian manifoldsrdquo Rendiconti di Matematica e delle sueApplicazioni VII vol 7 no 11 pp 377ndash396 1991

[23] M Falcitelli S Ianus and A M Pastore Riemannian Submer-sions and Related Topics World Scientific River Edge NJ USA2004

[24] E G Rio and D N Kupeli Semi-Riemannian Maps andTheir Applications Kluwer Academic Publisher DordrechtTheNetherlands 1999

[25] B Sahin ldquoSemi-invariant submersions from almost HermitianmanifoldsrdquoCanadianMathematical Bulletin vol 54 no 3 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 7: Research Article Paracomplex Paracontact Pseudo-Riemannian …downloads.hindawi.com/archive/2014/616487.pdf · 2019-07-31 · Research Article Paracomplex Paracontact Pseudo-Riemannian

Geometry 7

dimensional fibre submanifold 119891minus1(119902) of119872 119902 isin 119872 We get

119869119880 = 119886119869119880|DV + 119887119869

2

120585ℎ

= 119886119869119880|DV + 119887120585

ℎ

isinV oplus 120585ℎ

997904rArr 119869V subV oplus 120585ℎ

(66)

Again let 119881 isinV oplus 120585ℎ

Then 119881 = 119886119881|DV + 119887119869120585

ℎ

+ 119888120585ℎ

where

120578V(119881|DV ) = 0 D

V= ker 120578V 119886119881

|DV + 119887119869120585

ℎ

isin V and 119886 119887 119888 isin119862infin

(119872) We have

119869119881 = 119886119869119881|DV + 119887120585

ℎ

+ 119888119869120585ℎ

= (119886119869119881|DV + 119888119869120585

ℎ

)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

isinV

+ 119887120585ℎ

⏟⏟⏟⏟⏟⏟⏟

isin120585

ℎ

isinV oplus 120585ℎ

(67)

Now by (39) we get

119891lowast119869119881 = 120601 (119891

lowast119881) + 120578 (119891

lowast(119881)) 120585

= 119888 120601 (119891lowast120585ℎ

) + 120578 (120585) 120585

= 119888120585 isin 120585 sube 119869V

(68)

We get 119869119881 notinVHence 119869V subV oplus 120585

ℎ

that is 119869V sub DVoplus 119869120585ℎ

oplus 120585ℎ

(ii) Let 119883 = 119886119883

|

Dℎ+ 119887120585ℎ

isin H where H = Dℎ

oplus

120585ℎ

ker 120578ℎ = Dℎ

and 119886 119887 isin 119862infin(119872) Then

119869119883 = 119886119869119883|

Dℎ+ 119887119869120585ℎ

isinH oplus 119869120585ℎ

(69)

which implies that 119869H subH oplus 119869120585ℎ

Again let119884 isinHoplus119869120585

ℎ

Then119884 = 119886119884|

Dℎ+119887120585ℎ

+119888119869120585ℎ

notinHfor 119886 119887 119888 isin 119862infin(119872) We have

119869 119884 = 119886119869119884|

Dℎ+ 119887119869120585ℎ

+ 1198881198692

120585ℎ

= 119886119869119884|

Dℎ+ 119887119869120585ℎ

+ 119888120585ℎ

= 119885 + 119887119869120585ℎ

isinH oplus 119869120585ℎ

for some 119885 = 119886119869119884|

Dℎ+ 119888120585ℎ

isinH

(70)

We obtain 119869 119884 notinHHence 119869H sub H oplus 119869(120585

ℎ

) that is 119869H sub Dℎ

oplus 120585ℎ

oplus

119869120585ℎ

Example 13 Let (R63 119869 119892) be an almost paracomplex pseudo-

Riemannian manifold and let (R31 120601 120585 120578 119892) be an almost

paracontact pseudo-Riemannian manifold Consider a sub-mersion 119891 R6

3 (1199091 1199092 1199093 1199101 1199102 1199103)119905

rarr R31 (119906 V 119908)119905

defined by

119891 ((1199091 1199092 1199093 1199101 1199102 1199103)119905

) 997891997888rarr (1199091+ 1199092

radic2

1199101+ 1199102

radic2

1199103)

119905

(71)

Then the kernel of 119891lowastis

V = ker119891lowast

= Span1198811=

120597

1205971199091

minus120597

1205971199092

1198812=

120597

1205971199101

minus120597

1205971199102

120585V=

120597

1205971199093

(72)

which is the vertical distribution and the restriction of 119892 tothe fibres of 119891 is nondegenerate

The horizontal distribution is

H = (ker119891lowast)perp

= Span1198831=

120597

1205971199091

+120597

1205971199092

1198832=

120597

1205971199101

+120597

1205971199102

120585ℎ

=120597

1205971199103

(73)

The characteristic vector field 120585 = 120597120597119908 on R31has unique

horizontal lift 120585ℎ

which is the characteristic vector field onthe horizontal distributionH of R6

3

We also have

119892 (1198831 1198831) = 119892 (119891

lowast1198831 119891lowast1198831) = minus2

119892 (1198832 1198832) = 119892 (119891

lowast1198832 119891lowast1198832) = 2

119892 (120585ℎ

120585ℎ

) = 119892(119891lowast120585ℎ

119891lowast120585ℎ

) = 119892 (120585 120585) = 1

(74)

Thus the smooth map 119891 is a pseudo-Riemannian submer-sion

Also we obtain that there exists a 1-form 120578 = 1198891199093on R63

such that 120578(119869120585ℎ

) = 1 120578(120585ℎ

) = 0 and the map 119891 satisfies

119891lowast1198691198831= 120601119891lowast1198831+ 120578 (119883

1) 120585

119891lowast1198691198832= 120601119891lowast1198832+ 120578 (119883

2) 120585

119891lowast119869120585ℎ

= 120601119891lowast120585ℎ

+ 120578 (120585ℎ

) 120585

(75)

Hence the map 119891 is a paracomplex paracontact pseudo-Riemannian submersion from R6

3onto R3

1

Moreover we observe that for this submersion119891 we have

119869V subV oplus 120585ℎ

119869H subH oplus 119869120585ℎ

(76)

which verifies Proposition 12

8 Geometry

Proposition 14 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion and let the fibresof 119891 be pseudo-Riemannian submanifolds of 119872 Let 119883 119884 bebasic vector fields 119891-related to 119883 119884 respectively Let 119865 and Φbe the second fundamental forms and let nabla and nabla be the Levi-Civita connection on the total manifold119872 and base manifold119872 respectively Then one has

(i) 119891lowast((nabla119883119869)119884) = (nabla

119883120601)119884 + 120576119892(119884 nabla

119883120585)120585 + 120578(119884)nabla

119883120585

(ii) 119865 = 119891lowastlowastΦ + 120576120578 otimes 120578

(iii) 119891lowast((nabla119883119865)(119884 119885)) = (nabla

119883Φ)(119884 119885) + 120578(119884)119892(119885 nabla

119883120585) +

120578(119885)119892(119884 nabla119883120585)

Proof (i) In view of Definition 2 and Proposition 11 we have

119891lowast((nabla119883119869) 119884) = 119891

lowast(nabla119883(119869 119884) minus 119869 (nabla

119883119884))

= nabla119883(119891lowast(119869 119884)) minus 119891

lowast(119869 (nabla119883119884))

= nabla119883(120601119884) + nabla

119883(120578 (119884) 120585) minus 120601 (nabla

119883119884)

minus 120578 (nabla119883119884) 120585

= (nabla119883120601)119884 + nabla

119883(120576119892 (119884 120585) 120585) minus 120578 (nabla

119883119884) 120585

= (nabla119883120601)119884 + 120576119892 (nabla

119883119884 120585) 120585 + 120576119892 (119884 nabla

119883120585) 120585

+ 120576119892 (119884 120585) nabla119883120585 minus 120576119892 (nabla

119883119884 120585) 120585

= (nabla119883120601)119884 + 120576119892 (119884 nabla

119883120585) 120585 + 120578 (119884) nabla

119883120585

(77)

(ii) Since119891lowastlowastΦ is pullback ofΦ through the linearmap119891

lowast

we get

119891lowast

lowastΦ(119883 119884) = Φ (119883 119884) ∘ 119891 = 119892 (119883 120601119884) ∘ 119891

= 119892 (119883 119869 119884) minus 120576120578 (119884) 120578 (119883)

= 119865 (119883 119884) minus 120576120578 (119883) 120578 (119884)

(78)

which implies 119865 = 119891lowastlowastΦ + 120576120578 otimes 120578

(iii) By (23) we have

119891lowast((nabla119883119865) (119884 119885)) = 119892 (119891

lowast(119884) 119891

lowast((nabla119883119869)119885)) (79)

Now using (i) in the above equation we get (iii)

Theorem 15 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion and let the fibresof 119891 be pseudo-Riemannian submanifolds of 119872 Let 119883 119884 bebasic vector fields 119891-related to 119883 119884 respectively If the totalspace is para-Hermitian manifold then the almost paracontactstructure of base space is normal

Moreover if the almost paracontact structure of base spaceis normal then the Nijenhuis tensor of total space is vertical

Proof The Nijenhuis tensors 119873119869and 119873

120601of almost para-

complex structure 119869 and almost paracontact structure 120601 arerespectively defined by (8) and (11)

Using Definition 2 and properties of Sections 21 and 22we get the following identity

119891lowast(119873119869(119883 119884)) = 119873

(1)

(119883 119884) + 2119889120578 (120601119883 119884) 120585

minus 2119889120578 (120601119884119883) 120585

+ 2120578 (119883) 119889120578 (120585 119884) 120585 minus 2120578 (119884) 119889120578 (120585 119883) 120585

minus 120578 (119884)119873(3)

(119883) + 120578 (119883)119873(3)

(119884)

(80)

Using (12) (13) (14) and (15) (80) reduces to

119891lowast(119873119869(119883 119884)) = 119873

(1)

(119883 119884) + 119873(2)

(119883 119884) 120585

+ 120578 (119883)119873(4)

(119884) 120585

minus 120578 (119884)119873(4)

(119883) 120585 minus 120578 (119884)119873(3)

(119883)

+ 120578 (119883)119873(3)

(119884)

(81)

Since 119873119869(119883 119884) = 0 it follows from (81) that

tensors 119873(1) 119873(2) 119873(3) and 119873(4) vanish togetherHence the almost paracontact structure of base space is

normalConversely let the almost paracontact structure of the

base space be normalThen (81) implies that 119891

lowast(119873119869(119883 119884)) = 0

Hence119873119869(119883 119884) is vertical

Corollary 16 Let119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion and let the fibres of 119891 bepseudo-Riemannian submanifolds of 119872 Let 119883 119884 be basicvector fields 119891-related to 119883 119884 respectively Let the total spacebe para-Hermitian manifold and119873(1) vanishes Then the basespace is paracontact pseudometric manifold if and only if 120585 iskilling

Proof Let the total space be para-Hermitian and 119873(1) van-ishes Then from (80) we have

0 = 2119889120578 (120601119883 119884) 120585 minus 2119889120578 (120601119884119883) 120585 + 2120578 (119883) 119889120578 (120585 119884) 120585

minus 2120578 (119884) 119889120578 (120585 119883) 120585 minus 120578 (119884) (L120585120601)119883 + 120578 (119883) (L

120585120601)119884

(82)

If 120585 is killing then we have L120585120601 = 0 It immediately follows

from (82) that

119889120578 (120601119883 119884) minus 120578 (120601119884119883) + 120578 (119883) 119889120578 (120585 119884)

minus 120578 (119884) 119889120578 (120585 119883) = 0

(83)

In view of (6) and (7) the above equation gives 119889120578 = ΦConversely let the base space be paracontact Then 119889120578 =

ΦUsing (6) (7) and (82) we getL

120585120601 = 0

Hence the characteristic vector field 120585 is killing

Geometry 9

Theorem 17 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion and let the fibres of 119891 bepseudo-Riemannian submanifolds of 119872 Let 119883 119884 be basicvector fields 119891-related to 119883 119884 respectively If the total spaceis para-Kahler then the base space is paracosymplectic Theconverse is true if nabla

119883119869 is vertical

Proof We have for any 119883119884 isin Γ(119879119872) (nabla119883120601)119884 = 0 which

gives 119892(119885 (nabla119883120601)119884) = 0 for any 119885 isin Γ(119879119872)

From Proposition 14 we have

119892 (119885 (nabla119883120601)119884) + 120576120578 (119884) (nabla

119883120578)119885 + 120576120578 (119885) (nabla

119883120578) 119884 ∘ 119891

= 119892 (119885 (nabla119883119869) 119884)

(84)

Let nabla 119869 = 0 that is the total space is para-KahlerThen from(84) we obtain nabla120601 = 0 and nabla120578 = 0 Hence the base space isparacosymplectic

Again let (nabla119883120601)119884 = 0 and nabla120578 = 0 Then 119892(119885 (nabla

119883119869)119884) =

0 which implies that (nabla119883119869)119884 is a vertical vector field

Theorem 18 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion and let the fibres of 119891 bepseudo-Riemannian submanifolds of 119872 Let 119883 119884 and 119885 bebasic vector fields 119891-related to 119883 119884 and 119885 respectively If120578(119883)119885(120578(119884))+120578(119884)119883(120578(119885))minus120578(119883)119884(120578(119885)) = 0 then the totalspace is almost para-Kahler if and only if the base space119872 isan almost paracosymplectic manifold

Proof We have the following equation

3119889119865 (119883 119884 119885)

= 3 (119891lowast

lowast119889Φ) (119883 119884 119885) + 2120576120578 (119885) 119889120578 (119883 119884)

minus 2120576120578 (119884) 119889120578 (119883 119885)

+ 2120576120578 (119883) 119889120578 (119884 119885) + 2120576120578 (119883)119885 (120578 (119884))

+ 2120576120578 (119884)119883 (120578 (119885))

minus 2120576120578 (119883)119884 (120578 (119885))

(85)

If 119889120578 = 0 119889Φ = 0 and 120578(119883)119885(120578(119884)) + 120578(119884)119883(120578(119885)) minus

120578(119883)119884(120578(119885)) = 0 then from (85) we have 119889119865 = 0 Hencethe total space is almost para-Kahler

Conversely let 119889119865 = 0 and 120578(119883)119885(120578(119884))+120578(119884)119883(120578(119885))minus120578(119883)119884(120578(119885)) = 0

By using the above equation in (85) we have 119889120578 = 0 and119889Φ = 0

Hence the base space is almost paracosymplectic

Nowwe investigate the properties of fundamental tensorsT andA of a pseudo-Riemannian submersion

Lemma 19 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometric manifold119872 and

let the fibres of 119891 be pseudo-Riemannian submanifolds of 119872Then for any horizontal vector fields119883 119884 and for any verticalvector fields 119880 119881 on119872 one has

(i) A119883(119869 119884) = 119869(A

119883119884)

(ii) A119869119883(119884) = 119869(A

119883119884)

(iii) T119880(119869 119881) = 119869(T

119880119881)

(iv) T119869119880119881 = 119869(T

119880119881)

Proof The proof follows using similar steps as in Lemmas 3and 4 of [13] so we omit it

Lemma 20 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometric manifold119872 andlet the fibres of 119891 be pseudo-Riemannian submanifolds of 119872Then for any vector fields 119864 119865 on119872 one has

(i) A119864(119869119865) = 119869(A

119864119865)

(ii) T119864(119869119865) = 119869(T

119864119865)

Proof The proof follows from (37) and (38)

Theorem 21 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometric manifold119872 andlet the fibres of 119891 be pseudo-Riemannian submanifolds of 119872Then the horizontal distribution is integrable

Proof For any vertical vector field 119880 we have

119892 (119869 (A119883119884) 119880) = 119892 (A

119883119869 119884119880)

= minus119892 (119869119884A119883119880)

= minus119892 (119869119884 ℎ (nabla119880119883))

= 119892 (119884 ℎ (119869 (nabla119880119883)))

= 119892 (119884 ℎ (minusnabla119880119869)119883

+nabla119880(119869119883))

= 119892 (119884 ℎ nabla119880(119869119883)) = 119892 (119884A

119869119883119880)

= minus119892 (A119869119883119884119880) = minus119892 (119869 (A

119883119884) 119880)

(86)

Thus 119892(119869(A119883119884) 119880) = 0 which is true for all119883 and 119884

SoA119883119884 = 0

Hence the horizontal distribution is integrable

Theorem 22 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometric manifold119872 andlet the fibres of 119891 be pseudo-Riemannian submanifolds of 119872Then the submersion is an affine map onH

10 Geometry

Proof The second fundamental form of 119891 is defined by

(nabla119891lowast) (119864 119865) = (nabla

119891

119864119891lowast(119865)) ∘ 119891 minus 119891

lowast(nabla119864119865) (87)

where 119864 119865 isin Γ(119879119872) and nabla119891 is pullback connection of Levi-Civita connection nabla on119872 with respect to 119891

We have for any119883119884 isinH

(nabla119891lowast) (119883 119884) = (nabla

119891

119883

119891lowast(119884)) ∘ 119891 minus 119891

lowast(nabla119883119884) (88)

By using Lemma 1 we have 119891lowast(ℎ(nabla119883119884)) = (nabla

119883119884) ∘ 119891 which

implies nabla119891lowast= 0

Hence the submersion 119891 is an affine map onH

Theorem 23 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Hermitian man-ifold119872 onto an almost paracontact pseudometric manifold119872and let the fibres of 119891 be pseudo-Riemannian submanifolds of119872 Then the submersion is an affine map on V if and only ifthe fibres of 119891 are totally geodesic

Proof We have for any 119880119881 isinV

(nabla119891lowast) (119880 119881) = minus119891

lowast(ℎ (nabla119880119881)) (89)

which in view of (27) gives

(nabla119891lowast) (119880 119881) = minus119891

lowast(T119880119881) (90)

Let the fibres of 119891 be totally geodesic Then T = 0Consequently from the above equation we have nabla119891

lowast= 0

Thus the map 119891 is affine onVConversely let the submersion 119891 be an affine map onV

Then nabla119891lowast= 0 which impliesT = 0

Hence the fibres of 119891 are totally geodesic

Theorem 24 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Hermitian man-ifold119872 onto an almost paracontact pseudometric manifold119872and let the fibres of 119891 be pseudo-Riemannian submanifoldsof 119872 Then the submersion is an affine map if and only ifℎ(nabla119864ℎ119865)+A

ℎ119864V119865+TV119864V119865 is 119891-related to nabla

119883119884 for any 119864 119865 isin

Γ(119879119872)

Proof For any 119864 119865 isin Γ(119879119872) with 119891lowastℎ119864 = 119883 ∘ 119891 and 119891

lowastV119865 =

119884 ∘ 119891 we have

(nabla119891lowast) (119864 119865) = (nabla

119891lowastℎ119864(119891lowastℎ119865)) ∘ 119891 minus 119891

lowast(ℎ (nabla119864119865))

= (nabla119883119884) ∘ 119891 minus 119891

lowast(ℎ (nablaℎ119864ℎ119865 + nabla

ℎ119864V119865

+nablaV119864ℎ119865 + nablaV119864ℎ119865))

(91)

By using (27) and (31) in the above equation we have

(nabla119891lowast) (119864 119865) = (nabla

119883119884) ∘ 119891 minus 119891

lowast(ℎ (nabla119864ℎ119865) +A

ℎ119864V119865

+TV119864V119865) (92)

Let the submersion map be affine Then for any 119864 119865 isin

Γ(119879119872) (nabla119891lowast)(119864 119865) = 0 Equation (92) implies (nabla

119883119884) ∘ 119891 =

119891lowast(ℎ(nabla119864ℎ119865) +A

ℎ119864V119865 +TV119864ℎ119865)

Conversely let ℎ(nabla119864119865) + A

ℎ119864V119865 + TV119864ℎ119865 be 119891-related

to nabla119883119884 for any 119864 119865 isin Γ(119879119872) Then from (92) we have

(nabla119891lowast)(119864 119865) = 0Hence the submersion map 119891 is affine

4 Curvature Properties

In this section the paraholomorphic bisectional curvaturesand paraholomorphic sectional curvatures of total mani-fold base manifold and fibres of paracomplex paracontactpseudo-Riemannian submersion and their curvature proper-ties are studied

Let 119891 119872 rarr 119872 be a paracomplex paracontact pseudo-Riemannian submersion from an almost para-Hermitianmanifold (119872 119869 119892) onto an almost paracontact pseudometricmanifold (119872 120601 120585 120578 119892)

Suppose that the vector fields 119864 119865 span the 2-dimensional plane at point 119901 of 119872 and let R be theRiemannian curvature tensor of 119872 The paraholomorphicbisectional curvature 119861(119864 119865) of 119872 for any pair of nonzeronon-lightlike vector fields 119864 119865 on 119872 is defined by theformula

119861 (119864 119865) =

R (119864 119869119864 119865 119869119865)

119892 (119864 119864) 119892 (119865 119865) (93)

For a nonzero non-lightlike vector field 119864 the vector field119869119864 is also non-lightlike and 119864 119869119864 span the 2-dimensionalplane Then the paraholomorphic sectional curvature 119867(119864)is defined as

119867(119864) = 119861 (119864 119864) =

R (119864 119869119864 119864 119869119864)

119892 (119864 119864) 119892 (119864 119864) (94)

The curvature properties of Riemannian submersion andsemi-Riemannian submersion have been extensively studiedin the work of OrsquoNeill [1] and Gray [3]

Let 119861ℎ and 119861V be the paraholomorphic bisectional cur-vatures of horizontal and vertical spaces respectively Let119867ℎ and 119867V be the paraholomorphic sectional curvatures of

horizontal and vertical spaces respectively Let 119861 and 119867 bethe paraholomorphic bisectional and sectional curvatures ofthe base manifold respectively

Proposition 25 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion from an almostpara-Hermitian manifold onto an almost paracontact pseudo-metric manifold and let the fibres of 119891 be pseudo-Riemanniansubmanifolds of 119872 Let 119880 119881 be non-lightlike unit verticalvector fields and let119883 119884 be non-lightlike unit horizontal vectorfields on119872 Then one has

119861 (119880119881) = 119861V(119880 119881) + 119892 (T

119880(119869119881) T

119869119880119881)

minus 119892 (T119869119880(119869119881) T

119880119881)

(95)

Geometry 11

119861 (119883119880) = 119892 ((nabla119880A)119883

119869119883 119869119880) minus 119892 (A119883119869119880A

119869119883119880)

+ 119892 (A119883119880A119869119883119869119880) minus 119892 ((nabla

119869119880A)119883

119869119883119880)

+ 119892 (T119869119880119883T119880(119869119883))

minus 119892 (T119880119883T119869119880(119869119883))

(96)

119861 (119883 119884) = 119861ℎ

(119883 119884) minus 2119892 (A119883(119869119883) A

119884(119869 119884))

+ 119892 (A119869119883119884A119883(119869 119884))

minus 119892 (A119883119884A119869119883(119869 119884))

(97)

Proof Using Definitions (93) and (94) of paraholomorphicsectional curvature and fundamental equations of submer-sion obtained by OrsquoNeill [1] we have (95) (96) and (97)

Corollary 26 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion from an almostpara-Hermitian manifold onto an almost paracontact pseudo-metric manifold If the fibres of 119891 are totally geodesic pseudo-Riemannian submanifolds of119872 then for any non-lightlike unitvertical vector fields 119880 and 119881 one has

119861 (119880119881) = 119861V(119880 119881) (98)

Corollary 27 Let 119891 119872 rarr 119872 be a paracomplex para-contact pseudo-Riemannian submersion from an almost para-Hermitian manifold onto an almost paracontact pseudometricmanifold and let the fibres of 119891 be totally geodesic pseudo-Riemannian submanifolds of119872 If the horizontal distributionis integrable then for any non-lightlike unit horizontal vectorfields119883 and 119884 one has

119861 (119883 119884) = 119861ℎ

(119883 119884) (99)

Proposition 28 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion from an almostpara-Hermitian manifold onto an almost paracontact pseudo-metric manifold and let the fibres of 119891 be pseudo-Riemanniansubmanifolds of119872 Let 119880 and 119883 be non-lightlike unit verticalvector field and non-lightlike unit horizontal vector fieldrespectively Then one has

119867(119880) = 119867V(119880) +

10038171003817100381710038171003817T119880(119869119880)

10038171003817100381710038171003817

2

minus 119892 (T119869119880(119869119880) T

119880119880)

(100)

119867(119883) = 119867 (119883) ∘ 119891 minus 310038171003817100381710038171003817A119883(119869119883)

10038171003817100381710038171003817

2

(101)

Proof The proof is straightforward If we take 119880 = 119881 in (95)and119883 = 119884 in (97) we have (98) and (99)

Corollary 29 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion from an almostpara-Hermitian manifold onto an almost paracontact pseudo-metric manifold If the fibres of 119891 are totally geodesic pseudo-Riemannian submanifolds of 119872 then the total manifold andfibres of119891have the same paraholomorphic sectional curvatures

Proof Since the fibres are totally geodesic T = 0 conse-quently we have

119867(119880) = 119867V(119880) (102)

Corollary 30 Let 119891 119872 rarr 119872 be a paracomplex para-contact pseudo-Riemannian submersion from an almost para-Hermitian manifold onto an almost paracontact pseudometricmanifold and let the fibres of 119891 be totally geodesic pseudo-Riemannian submanifolds of119872 If the horizontal distributionis integrable then the base manifold and horizontal distribu-tion have the same paraholomorphic sectional curvatures

Proof Since the horizontal distribution is integrableA = 0consequently we have

119867(119883) = 119867 (119883) ∘ 119891 (103)

Theorem 31 Let 119891 119872119898

rarr 119872119899 be a paracomplex

paracontact pseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometricmanifold 119872 and let the fibres of 119891 be pseudo-Riemanniansubmanifolds of 119872 If 119880 119881 are the non-lightlike unit verticalvector fields and 119883 119884 are the non-lightlike unit horizontalvector fields then one has

119861 (119880119881) = 119861V(119880 119881) (104)

119861 (119883119880) = minus210038171003817100381710038171003817T11988011988310038171003817100381710038171003817

2

(105)

119861 (119883 119884) = 119861 (119883 119884) ∘ 119891 (106)

Proof Using results of Lemma 19 in (95) we have

119861 (119880119881) = 119861V(119880 119881) minus 119892 (119869 (T

119880119881) 119869 (T

119880119881))

minus 119892 (1198692

(T119880119881) T

119880119881)

= 119861 (119880119881) + 119892 (T119880119881T119880119881) minus 119892 (T

119880119881T119880119881)

= 119861V(119880 119881)

(107)

Applying results of Lemma 19 in (96) we have

119861 (119883119880) = 119892 ((nabla119880A)119883

(119869119883) 119869119880) minus 119892 ((nabla119869119880A)119883

(119869119883) 119880)

+ 21003817100381710038171003817A119883119880

1003817100381710038171003817

2

minus 210038171003817100381710038171003817T11988011988310038171003817100381710038171003817

2

(108)

Since byTheorem 21 the horizontal distribution is integrablewe haveA = 0 which implies

119861 (119883119880) = minus210038171003817100381710038171003817T11988011988310038171003817100381710038171003817

2

(109)

In view ofA = 0 (104) follows from (97)

12 Geometry

Theorem 32 Let 119891 119872119898

rarr 119872119899 be a paracomplex

paracontact pseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometricmanifold 119872 and let the fibres of 119891 be pseudo-Riemanniansubmanifolds of119872 If 119880 119883 are non-lightlike unit vertical andnon-lightlike unit horizontal vector fields respectively then onehas

119867(119880) = 119867V(119880) minus 2

1003817100381710038171003817T1198801198801003817100381710038171003817

2

(110)

119867(119883) = 119867 (119883) ∘ 119891 (111)

Proof Since 119891 is the paracomplex paracontact pseudo-Riemannian submersion from a para-Kahler manifold 119872

onto an almost paracontact pseudometric manifold 119872 by(16) and equations of Lemma 19 we have

119892 (T119869119880(119869119880) T

119880119880) = 119892 (119869

2

(T119880119880) T

119880119880) =

1003817100381710038171003817T1198801198801003817100381710038171003817

2

119892 (T119880(119869119880) T

119880(119869119880)) = minus119892 (T

119880119880T119880119880) = minus

1003817100381710038171003817T1198801198801003817100381710038171003817

2

(112)

and by using the above results in (100) we have

119867(119880) = 119867V(119880) minus

1003817100381710038171003817T1198801198801003817100381710038171003817

2

minus1003817100381710038171003817T119880119880

1003817100381710038171003817

2

= 119867V(119880) minus 2

1003817100381710038171003817T1198801198801003817100381710038171003817

2

(113)

Again since horizontal distribution is integrable we haveA = 0 and putting it in (101) we obtain (111)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Uma Shankar Verma is thankful to University Grant Com-mission New Delhi India for financial support

References

[1] B OrsquoNeill ldquoThe fundamental equations of a submersionrdquo TheMichigan Mathematical Journal vol 13 pp 459ndash469 1966

[2] B OrsquoNeill Semi-Riemannian Geometry with Applications toRelativity vol 103 of Pure and Applied Mathematics AcademicPress New York NY USA 1983

[3] A Gray ldquoPseudo-Riemannian almost product manifolds andsubmersionsrdquo vol 16 pp 715ndash737 1967

[4] J P Bourguignon and H B Lawson A Mathematicians Visitto Kaluza-Klein Theory Rendiconti del Seminario Matematico1989

[5] S Ianus and M Visinescu ldquoSpace-time compactification andRiemannian submersionsrdquo in The Mathematical Heritage GRassias and C F Gauss Eds pp 358ndash371 World ScientificRiver Edge NJ USA 1991

[6] J P Bourguignon and H B Lawson Jr ldquoStability and isolationphenomena for Yang-Mills fieldsrdquo Communications in Mathe-matical Physics vol 79 no 2 pp 189ndash230 1981

[7] B Watson ldquoG 1198661015840

-Riemannian submersions and nonlineargauge field equations of general relativityrdquo in Global AnalysismdashAnalysis on Manifolds T Rassias and M Morse Eds vol 57 ofTeubner-Texte zur Mathematik pp 324ndash349 Teubner LeipzigGermany 1983

[8] C Altafini ldquoRedundant robotic chains on Riemannian submer-sionsrdquo IEEE Transactions on Robotics and Automation vol 20no 2 pp 335ndash340 2004

[9] M T Mustafa ldquoApplications of harmonic morphisms to grav-ityrdquo Journal of Mathematical Physics vol 41 no 10 pp 6918ndash6929 2000

[10] B Watson ldquoAlmost Hermitian submersionsrdquo Journal of Differ-ential Geometry vol 11 no 1 pp 147ndash165 1976

[11] D Chinea ldquoAlmost contact metric submersionsrdquo Rendiconti delCircolo Matematico di Palermo II vol 34 no 1 pp 319ndash3301984

[12] D Chinea ldquoTransference of structures on almost complexcontact metric submersionsrdquo Houston Journal of Mathematicsvol 14 no 1 pp 9ndash22 1988

[13] Y Gunduzalp and B Sahin ldquoPara-contact para-complexpseudo-Riemannian submersionsrdquo Bulletin of the MalaysianMathematical Sciences Society

[14] I Sato ldquoOn a structure similar to the almost contact structurerdquoTensor vol 30 no 3 pp 219ndash224 1976

[15] P K Rashevskij ldquoThe scalar field in a stratified spacerdquoTrudy Seminara po Vektornomu i Tenzornomu Analizu s ikhPrilozheniyami k Geometrii Mekhanike i Fizike vol 6 pp 225ndash248 1948

[16] P Libermann ldquoSur les structures presque paracomplexesrdquoComptes Rendus de lrsquoAcademie des Sciences I vol 234 pp 2517ndash2519 1952

[17] E M Patterson ldquoRiemann extensions which have Kahlermetricsrdquo Proceedings of the Royal Society of Edinburgh AMathematics vol 64 pp 113ndash126 1954

[18] S Sasaki ldquoOn differentiable manifolds with certain structureswhich are closely related to almost contact structure Irdquo TheTohoku Mathematical Journal vol 12 pp 459ndash476 1960

[19] S Zamkovoy ldquoCanonical connections on paracontact mani-foldsrdquo Annals of Global Analysis and Geometry vol 36 no 1pp 37ndash60 2009

[20] D E Blair Riemannian Geometry of Contact and SymplecticManifolds vol 23 of Progress in Mathematics BirkhauserBoston Mass USA 2002

[21] V Cruceanu P Fortuny and P M Gadea ldquoA survey onparacomplex geometryrdquoThe Rocky Mountain Journal of Mathe-matics vol 26 no 1 pp 83ndash115 1996

[22] P M Gadea and J M Masque ldquoClassification of almost para-Hermitian manifoldsrdquo Rendiconti di Matematica e delle sueApplicazioni VII vol 7 no 11 pp 377ndash396 1991

[23] M Falcitelli S Ianus and A M Pastore Riemannian Submer-sions and Related Topics World Scientific River Edge NJ USA2004

[24] E G Rio and D N Kupeli Semi-Riemannian Maps andTheir Applications Kluwer Academic Publisher DordrechtTheNetherlands 1999

[25] B Sahin ldquoSemi-invariant submersions from almost HermitianmanifoldsrdquoCanadianMathematical Bulletin vol 54 no 3 2011

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Stochastic AnalysisInternational Journal of

Page 8: Research Article Paracomplex Paracontact Pseudo-Riemannian …downloads.hindawi.com/archive/2014/616487.pdf · 2019-07-31 · Research Article Paracomplex Paracontact Pseudo-Riemannian

8 Geometry

Proposition 14 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion and let the fibresof 119891 be pseudo-Riemannian submanifolds of 119872 Let 119883 119884 bebasic vector fields 119891-related to 119883 119884 respectively Let 119865 and Φbe the second fundamental forms and let nabla and nabla be the Levi-Civita connection on the total manifold119872 and base manifold119872 respectively Then one has

(i) 119891lowast((nabla119883119869)119884) = (nabla

119883120601)119884 + 120576119892(119884 nabla

119883120585)120585 + 120578(119884)nabla

119883120585

(ii) 119865 = 119891lowastlowastΦ + 120576120578 otimes 120578

(iii) 119891lowast((nabla119883119865)(119884 119885)) = (nabla

119883Φ)(119884 119885) + 120578(119884)119892(119885 nabla

119883120585) +

120578(119885)119892(119884 nabla119883120585)

Proof (i) In view of Definition 2 and Proposition 11 we have

119891lowast((nabla119883119869) 119884) = 119891

lowast(nabla119883(119869 119884) minus 119869 (nabla

119883119884))

= nabla119883(119891lowast(119869 119884)) minus 119891

lowast(119869 (nabla119883119884))

= nabla119883(120601119884) + nabla

119883(120578 (119884) 120585) minus 120601 (nabla

119883119884)

minus 120578 (nabla119883119884) 120585

= (nabla119883120601)119884 + nabla

119883(120576119892 (119884 120585) 120585) minus 120578 (nabla

119883119884) 120585

= (nabla119883120601)119884 + 120576119892 (nabla

119883119884 120585) 120585 + 120576119892 (119884 nabla

119883120585) 120585

+ 120576119892 (119884 120585) nabla119883120585 minus 120576119892 (nabla

119883119884 120585) 120585

= (nabla119883120601)119884 + 120576119892 (119884 nabla

119883120585) 120585 + 120578 (119884) nabla

119883120585

(77)

(ii) Since119891lowastlowastΦ is pullback ofΦ through the linearmap119891

lowast

we get

119891lowast

lowastΦ(119883 119884) = Φ (119883 119884) ∘ 119891 = 119892 (119883 120601119884) ∘ 119891

= 119892 (119883 119869 119884) minus 120576120578 (119884) 120578 (119883)

= 119865 (119883 119884) minus 120576120578 (119883) 120578 (119884)

(78)

which implies 119865 = 119891lowastlowastΦ + 120576120578 otimes 120578

(iii) By (23) we have

119891lowast((nabla119883119865) (119884 119885)) = 119892 (119891

lowast(119884) 119891

lowast((nabla119883119869)119885)) (79)

Now using (i) in the above equation we get (iii)

Theorem 15 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion and let the fibresof 119891 be pseudo-Riemannian submanifolds of 119872 Let 119883 119884 bebasic vector fields 119891-related to 119883 119884 respectively If the totalspace is para-Hermitian manifold then the almost paracontactstructure of base space is normal

Moreover if the almost paracontact structure of base spaceis normal then the Nijenhuis tensor of total space is vertical

Proof The Nijenhuis tensors 119873119869and 119873

120601of almost para-

complex structure 119869 and almost paracontact structure 120601 arerespectively defined by (8) and (11)

Using Definition 2 and properties of Sections 21 and 22we get the following identity

119891lowast(119873119869(119883 119884)) = 119873

(1)

(119883 119884) + 2119889120578 (120601119883 119884) 120585

minus 2119889120578 (120601119884119883) 120585

+ 2120578 (119883) 119889120578 (120585 119884) 120585 minus 2120578 (119884) 119889120578 (120585 119883) 120585

minus 120578 (119884)119873(3)

(119883) + 120578 (119883)119873(3)

(119884)

(80)

Using (12) (13) (14) and (15) (80) reduces to

119891lowast(119873119869(119883 119884)) = 119873

(1)

(119883 119884) + 119873(2)

(119883 119884) 120585

+ 120578 (119883)119873(4)

(119884) 120585

minus 120578 (119884)119873(4)

(119883) 120585 minus 120578 (119884)119873(3)

(119883)

+ 120578 (119883)119873(3)

(119884)

(81)

Since 119873119869(119883 119884) = 0 it follows from (81) that

tensors 119873(1) 119873(2) 119873(3) and 119873(4) vanish togetherHence the almost paracontact structure of base space is

normalConversely let the almost paracontact structure of the

base space be normalThen (81) implies that 119891

lowast(119873119869(119883 119884)) = 0

Hence119873119869(119883 119884) is vertical

Corollary 16 Let119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion and let the fibres of 119891 bepseudo-Riemannian submanifolds of 119872 Let 119883 119884 be basicvector fields 119891-related to 119883 119884 respectively Let the total spacebe para-Hermitian manifold and119873(1) vanishes Then the basespace is paracontact pseudometric manifold if and only if 120585 iskilling

Proof Let the total space be para-Hermitian and 119873(1) van-ishes Then from (80) we have

0 = 2119889120578 (120601119883 119884) 120585 minus 2119889120578 (120601119884119883) 120585 + 2120578 (119883) 119889120578 (120585 119884) 120585

minus 2120578 (119884) 119889120578 (120585 119883) 120585 minus 120578 (119884) (L120585120601)119883 + 120578 (119883) (L

120585120601)119884

(82)

If 120585 is killing then we have L120585120601 = 0 It immediately follows

from (82) that

119889120578 (120601119883 119884) minus 120578 (120601119884119883) + 120578 (119883) 119889120578 (120585 119884)

minus 120578 (119884) 119889120578 (120585 119883) = 0

(83)

In view of (6) and (7) the above equation gives 119889120578 = ΦConversely let the base space be paracontact Then 119889120578 =

ΦUsing (6) (7) and (82) we getL

120585120601 = 0

Hence the characteristic vector field 120585 is killing

Geometry 9

Theorem 17 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion and let the fibres of 119891 bepseudo-Riemannian submanifolds of 119872 Let 119883 119884 be basicvector fields 119891-related to 119883 119884 respectively If the total spaceis para-Kahler then the base space is paracosymplectic Theconverse is true if nabla

119883119869 is vertical

Proof We have for any 119883119884 isin Γ(119879119872) (nabla119883120601)119884 = 0 which

gives 119892(119885 (nabla119883120601)119884) = 0 for any 119885 isin Γ(119879119872)

From Proposition 14 we have

119892 (119885 (nabla119883120601)119884) + 120576120578 (119884) (nabla

119883120578)119885 + 120576120578 (119885) (nabla

119883120578) 119884 ∘ 119891

= 119892 (119885 (nabla119883119869) 119884)

(84)

Let nabla 119869 = 0 that is the total space is para-KahlerThen from(84) we obtain nabla120601 = 0 and nabla120578 = 0 Hence the base space isparacosymplectic

Again let (nabla119883120601)119884 = 0 and nabla120578 = 0 Then 119892(119885 (nabla

119883119869)119884) =

0 which implies that (nabla119883119869)119884 is a vertical vector field

Theorem 18 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion and let the fibres of 119891 bepseudo-Riemannian submanifolds of 119872 Let 119883 119884 and 119885 bebasic vector fields 119891-related to 119883 119884 and 119885 respectively If120578(119883)119885(120578(119884))+120578(119884)119883(120578(119885))minus120578(119883)119884(120578(119885)) = 0 then the totalspace is almost para-Kahler if and only if the base space119872 isan almost paracosymplectic manifold

Proof We have the following equation

3119889119865 (119883 119884 119885)

= 3 (119891lowast

lowast119889Φ) (119883 119884 119885) + 2120576120578 (119885) 119889120578 (119883 119884)

minus 2120576120578 (119884) 119889120578 (119883 119885)

+ 2120576120578 (119883) 119889120578 (119884 119885) + 2120576120578 (119883)119885 (120578 (119884))

+ 2120576120578 (119884)119883 (120578 (119885))

minus 2120576120578 (119883)119884 (120578 (119885))

(85)

If 119889120578 = 0 119889Φ = 0 and 120578(119883)119885(120578(119884)) + 120578(119884)119883(120578(119885)) minus

120578(119883)119884(120578(119885)) = 0 then from (85) we have 119889119865 = 0 Hencethe total space is almost para-Kahler

Conversely let 119889119865 = 0 and 120578(119883)119885(120578(119884))+120578(119884)119883(120578(119885))minus120578(119883)119884(120578(119885)) = 0

By using the above equation in (85) we have 119889120578 = 0 and119889Φ = 0

Hence the base space is almost paracosymplectic

Nowwe investigate the properties of fundamental tensorsT andA of a pseudo-Riemannian submersion

Lemma 19 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometric manifold119872 and

let the fibres of 119891 be pseudo-Riemannian submanifolds of 119872Then for any horizontal vector fields119883 119884 and for any verticalvector fields 119880 119881 on119872 one has

(i) A119883(119869 119884) = 119869(A

119883119884)

(ii) A119869119883(119884) = 119869(A

119883119884)

(iii) T119880(119869 119881) = 119869(T

119880119881)

(iv) T119869119880119881 = 119869(T

119880119881)

Proof The proof follows using similar steps as in Lemmas 3and 4 of [13] so we omit it

Lemma 20 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometric manifold119872 andlet the fibres of 119891 be pseudo-Riemannian submanifolds of 119872Then for any vector fields 119864 119865 on119872 one has

(i) A119864(119869119865) = 119869(A

119864119865)

(ii) T119864(119869119865) = 119869(T

119864119865)

Proof The proof follows from (37) and (38)

Theorem 21 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometric manifold119872 andlet the fibres of 119891 be pseudo-Riemannian submanifolds of 119872Then the horizontal distribution is integrable

Proof For any vertical vector field 119880 we have

119892 (119869 (A119883119884) 119880) = 119892 (A

119883119869 119884119880)

= minus119892 (119869119884A119883119880)

= minus119892 (119869119884 ℎ (nabla119880119883))

= 119892 (119884 ℎ (119869 (nabla119880119883)))

= 119892 (119884 ℎ (minusnabla119880119869)119883

+nabla119880(119869119883))

= 119892 (119884 ℎ nabla119880(119869119883)) = 119892 (119884A

119869119883119880)

= minus119892 (A119869119883119884119880) = minus119892 (119869 (A

119883119884) 119880)

(86)

Thus 119892(119869(A119883119884) 119880) = 0 which is true for all119883 and 119884

SoA119883119884 = 0

Hence the horizontal distribution is integrable

Theorem 22 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometric manifold119872 andlet the fibres of 119891 be pseudo-Riemannian submanifolds of 119872Then the submersion is an affine map onH

10 Geometry

Proof The second fundamental form of 119891 is defined by

(nabla119891lowast) (119864 119865) = (nabla

119891

119864119891lowast(119865)) ∘ 119891 minus 119891

lowast(nabla119864119865) (87)

where 119864 119865 isin Γ(119879119872) and nabla119891 is pullback connection of Levi-Civita connection nabla on119872 with respect to 119891

We have for any119883119884 isinH

(nabla119891lowast) (119883 119884) = (nabla

119891

119883

119891lowast(119884)) ∘ 119891 minus 119891

lowast(nabla119883119884) (88)

By using Lemma 1 we have 119891lowast(ℎ(nabla119883119884)) = (nabla

119883119884) ∘ 119891 which

implies nabla119891lowast= 0

Hence the submersion 119891 is an affine map onH

Theorem 23 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Hermitian man-ifold119872 onto an almost paracontact pseudometric manifold119872and let the fibres of 119891 be pseudo-Riemannian submanifolds of119872 Then the submersion is an affine map on V if and only ifthe fibres of 119891 are totally geodesic

Proof We have for any 119880119881 isinV

(nabla119891lowast) (119880 119881) = minus119891

lowast(ℎ (nabla119880119881)) (89)

which in view of (27) gives

(nabla119891lowast) (119880 119881) = minus119891

lowast(T119880119881) (90)

Let the fibres of 119891 be totally geodesic Then T = 0Consequently from the above equation we have nabla119891

lowast= 0

Thus the map 119891 is affine onVConversely let the submersion 119891 be an affine map onV

Then nabla119891lowast= 0 which impliesT = 0

Hence the fibres of 119891 are totally geodesic

Theorem 24 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Hermitian man-ifold119872 onto an almost paracontact pseudometric manifold119872and let the fibres of 119891 be pseudo-Riemannian submanifoldsof 119872 Then the submersion is an affine map if and only ifℎ(nabla119864ℎ119865)+A

ℎ119864V119865+TV119864V119865 is 119891-related to nabla

119883119884 for any 119864 119865 isin

Γ(119879119872)

Proof For any 119864 119865 isin Γ(119879119872) with 119891lowastℎ119864 = 119883 ∘ 119891 and 119891

lowastV119865 =

119884 ∘ 119891 we have

(nabla119891lowast) (119864 119865) = (nabla

119891lowastℎ119864(119891lowastℎ119865)) ∘ 119891 minus 119891

lowast(ℎ (nabla119864119865))

= (nabla119883119884) ∘ 119891 minus 119891

lowast(ℎ (nablaℎ119864ℎ119865 + nabla

ℎ119864V119865

+nablaV119864ℎ119865 + nablaV119864ℎ119865))

(91)

By using (27) and (31) in the above equation we have

(nabla119891lowast) (119864 119865) = (nabla

119883119884) ∘ 119891 minus 119891

lowast(ℎ (nabla119864ℎ119865) +A

ℎ119864V119865

+TV119864V119865) (92)

Let the submersion map be affine Then for any 119864 119865 isin

Γ(119879119872) (nabla119891lowast)(119864 119865) = 0 Equation (92) implies (nabla

119883119884) ∘ 119891 =

119891lowast(ℎ(nabla119864ℎ119865) +A

ℎ119864V119865 +TV119864ℎ119865)

Conversely let ℎ(nabla119864119865) + A

ℎ119864V119865 + TV119864ℎ119865 be 119891-related

to nabla119883119884 for any 119864 119865 isin Γ(119879119872) Then from (92) we have

(nabla119891lowast)(119864 119865) = 0Hence the submersion map 119891 is affine

4 Curvature Properties

In this section the paraholomorphic bisectional curvaturesand paraholomorphic sectional curvatures of total mani-fold base manifold and fibres of paracomplex paracontactpseudo-Riemannian submersion and their curvature proper-ties are studied

Let 119891 119872 rarr 119872 be a paracomplex paracontact pseudo-Riemannian submersion from an almost para-Hermitianmanifold (119872 119869 119892) onto an almost paracontact pseudometricmanifold (119872 120601 120585 120578 119892)

Suppose that the vector fields 119864 119865 span the 2-dimensional plane at point 119901 of 119872 and let R be theRiemannian curvature tensor of 119872 The paraholomorphicbisectional curvature 119861(119864 119865) of 119872 for any pair of nonzeronon-lightlike vector fields 119864 119865 on 119872 is defined by theformula

119861 (119864 119865) =

R (119864 119869119864 119865 119869119865)

119892 (119864 119864) 119892 (119865 119865) (93)

For a nonzero non-lightlike vector field 119864 the vector field119869119864 is also non-lightlike and 119864 119869119864 span the 2-dimensionalplane Then the paraholomorphic sectional curvature 119867(119864)is defined as

119867(119864) = 119861 (119864 119864) =

R (119864 119869119864 119864 119869119864)

119892 (119864 119864) 119892 (119864 119864) (94)

The curvature properties of Riemannian submersion andsemi-Riemannian submersion have been extensively studiedin the work of OrsquoNeill [1] and Gray [3]

Let 119861ℎ and 119861V be the paraholomorphic bisectional cur-vatures of horizontal and vertical spaces respectively Let119867ℎ and 119867V be the paraholomorphic sectional curvatures of

horizontal and vertical spaces respectively Let 119861 and 119867 bethe paraholomorphic bisectional and sectional curvatures ofthe base manifold respectively

Proposition 25 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion from an almostpara-Hermitian manifold onto an almost paracontact pseudo-metric manifold and let the fibres of 119891 be pseudo-Riemanniansubmanifolds of 119872 Let 119880 119881 be non-lightlike unit verticalvector fields and let119883 119884 be non-lightlike unit horizontal vectorfields on119872 Then one has

119861 (119880119881) = 119861V(119880 119881) + 119892 (T

119880(119869119881) T

119869119880119881)

minus 119892 (T119869119880(119869119881) T

119880119881)

(95)

Geometry 11

119861 (119883119880) = 119892 ((nabla119880A)119883

119869119883 119869119880) minus 119892 (A119883119869119880A

119869119883119880)

+ 119892 (A119883119880A119869119883119869119880) minus 119892 ((nabla

119869119880A)119883

119869119883119880)

+ 119892 (T119869119880119883T119880(119869119883))

minus 119892 (T119880119883T119869119880(119869119883))

(96)

119861 (119883 119884) = 119861ℎ

(119883 119884) minus 2119892 (A119883(119869119883) A

119884(119869 119884))

+ 119892 (A119869119883119884A119883(119869 119884))

minus 119892 (A119883119884A119869119883(119869 119884))

(97)

Proof Using Definitions (93) and (94) of paraholomorphicsectional curvature and fundamental equations of submer-sion obtained by OrsquoNeill [1] we have (95) (96) and (97)

Corollary 26 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion from an almostpara-Hermitian manifold onto an almost paracontact pseudo-metric manifold If the fibres of 119891 are totally geodesic pseudo-Riemannian submanifolds of119872 then for any non-lightlike unitvertical vector fields 119880 and 119881 one has

119861 (119880119881) = 119861V(119880 119881) (98)

Corollary 27 Let 119891 119872 rarr 119872 be a paracomplex para-contact pseudo-Riemannian submersion from an almost para-Hermitian manifold onto an almost paracontact pseudometricmanifold and let the fibres of 119891 be totally geodesic pseudo-Riemannian submanifolds of119872 If the horizontal distributionis integrable then for any non-lightlike unit horizontal vectorfields119883 and 119884 one has

119861 (119883 119884) = 119861ℎ

(119883 119884) (99)

Proposition 28 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion from an almostpara-Hermitian manifold onto an almost paracontact pseudo-metric manifold and let the fibres of 119891 be pseudo-Riemanniansubmanifolds of119872 Let 119880 and 119883 be non-lightlike unit verticalvector field and non-lightlike unit horizontal vector fieldrespectively Then one has

119867(119880) = 119867V(119880) +

10038171003817100381710038171003817T119880(119869119880)

10038171003817100381710038171003817

2

minus 119892 (T119869119880(119869119880) T

119880119880)

(100)

119867(119883) = 119867 (119883) ∘ 119891 minus 310038171003817100381710038171003817A119883(119869119883)

10038171003817100381710038171003817

2

(101)

Proof The proof is straightforward If we take 119880 = 119881 in (95)and119883 = 119884 in (97) we have (98) and (99)

Corollary 29 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion from an almostpara-Hermitian manifold onto an almost paracontact pseudo-metric manifold If the fibres of 119891 are totally geodesic pseudo-Riemannian submanifolds of 119872 then the total manifold andfibres of119891have the same paraholomorphic sectional curvatures

Proof Since the fibres are totally geodesic T = 0 conse-quently we have

119867(119880) = 119867V(119880) (102)

Corollary 30 Let 119891 119872 rarr 119872 be a paracomplex para-contact pseudo-Riemannian submersion from an almost para-Hermitian manifold onto an almost paracontact pseudometricmanifold and let the fibres of 119891 be totally geodesic pseudo-Riemannian submanifolds of119872 If the horizontal distributionis integrable then the base manifold and horizontal distribu-tion have the same paraholomorphic sectional curvatures

Proof Since the horizontal distribution is integrableA = 0consequently we have

119867(119883) = 119867 (119883) ∘ 119891 (103)

Theorem 31 Let 119891 119872119898

rarr 119872119899 be a paracomplex

paracontact pseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometricmanifold 119872 and let the fibres of 119891 be pseudo-Riemanniansubmanifolds of 119872 If 119880 119881 are the non-lightlike unit verticalvector fields and 119883 119884 are the non-lightlike unit horizontalvector fields then one has

119861 (119880119881) = 119861V(119880 119881) (104)

119861 (119883119880) = minus210038171003817100381710038171003817T11988011988310038171003817100381710038171003817

2

(105)

119861 (119883 119884) = 119861 (119883 119884) ∘ 119891 (106)

Proof Using results of Lemma 19 in (95) we have

119861 (119880119881) = 119861V(119880 119881) minus 119892 (119869 (T

119880119881) 119869 (T

119880119881))

minus 119892 (1198692

(T119880119881) T

119880119881)

= 119861 (119880119881) + 119892 (T119880119881T119880119881) minus 119892 (T

119880119881T119880119881)

= 119861V(119880 119881)

(107)

Applying results of Lemma 19 in (96) we have

119861 (119883119880) = 119892 ((nabla119880A)119883

(119869119883) 119869119880) minus 119892 ((nabla119869119880A)119883

(119869119883) 119880)

+ 21003817100381710038171003817A119883119880

1003817100381710038171003817

2

minus 210038171003817100381710038171003817T11988011988310038171003817100381710038171003817

2

(108)

Since byTheorem 21 the horizontal distribution is integrablewe haveA = 0 which implies

119861 (119883119880) = minus210038171003817100381710038171003817T11988011988310038171003817100381710038171003817

2

(109)

In view ofA = 0 (104) follows from (97)

12 Geometry

Theorem 32 Let 119891 119872119898

rarr 119872119899 be a paracomplex

paracontact pseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometricmanifold 119872 and let the fibres of 119891 be pseudo-Riemanniansubmanifolds of119872 If 119880 119883 are non-lightlike unit vertical andnon-lightlike unit horizontal vector fields respectively then onehas

119867(119880) = 119867V(119880) minus 2

1003817100381710038171003817T1198801198801003817100381710038171003817

2

(110)

119867(119883) = 119867 (119883) ∘ 119891 (111)

Proof Since 119891 is the paracomplex paracontact pseudo-Riemannian submersion from a para-Kahler manifold 119872

onto an almost paracontact pseudometric manifold 119872 by(16) and equations of Lemma 19 we have

119892 (T119869119880(119869119880) T

119880119880) = 119892 (119869

2

(T119880119880) T

119880119880) =

1003817100381710038171003817T1198801198801003817100381710038171003817

2

119892 (T119880(119869119880) T

119880(119869119880)) = minus119892 (T

119880119880T119880119880) = minus

1003817100381710038171003817T1198801198801003817100381710038171003817

2

(112)

and by using the above results in (100) we have

119867(119880) = 119867V(119880) minus

1003817100381710038171003817T1198801198801003817100381710038171003817

2

minus1003817100381710038171003817T119880119880

1003817100381710038171003817

2

= 119867V(119880) minus 2

1003817100381710038171003817T1198801198801003817100381710038171003817

2

(113)

Again since horizontal distribution is integrable we haveA = 0 and putting it in (101) we obtain (111)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Uma Shankar Verma is thankful to University Grant Com-mission New Delhi India for financial support

References

[1] B OrsquoNeill ldquoThe fundamental equations of a submersionrdquo TheMichigan Mathematical Journal vol 13 pp 459ndash469 1966

[2] B OrsquoNeill Semi-Riemannian Geometry with Applications toRelativity vol 103 of Pure and Applied Mathematics AcademicPress New York NY USA 1983

[3] A Gray ldquoPseudo-Riemannian almost product manifolds andsubmersionsrdquo vol 16 pp 715ndash737 1967

[4] J P Bourguignon and H B Lawson A Mathematicians Visitto Kaluza-Klein Theory Rendiconti del Seminario Matematico1989

[5] S Ianus and M Visinescu ldquoSpace-time compactification andRiemannian submersionsrdquo in The Mathematical Heritage GRassias and C F Gauss Eds pp 358ndash371 World ScientificRiver Edge NJ USA 1991

[6] J P Bourguignon and H B Lawson Jr ldquoStability and isolationphenomena for Yang-Mills fieldsrdquo Communications in Mathe-matical Physics vol 79 no 2 pp 189ndash230 1981

[7] B Watson ldquoG 1198661015840

-Riemannian submersions and nonlineargauge field equations of general relativityrdquo in Global AnalysismdashAnalysis on Manifolds T Rassias and M Morse Eds vol 57 ofTeubner-Texte zur Mathematik pp 324ndash349 Teubner LeipzigGermany 1983

[8] C Altafini ldquoRedundant robotic chains on Riemannian submer-sionsrdquo IEEE Transactions on Robotics and Automation vol 20no 2 pp 335ndash340 2004

[9] M T Mustafa ldquoApplications of harmonic morphisms to grav-ityrdquo Journal of Mathematical Physics vol 41 no 10 pp 6918ndash6929 2000

[10] B Watson ldquoAlmost Hermitian submersionsrdquo Journal of Differ-ential Geometry vol 11 no 1 pp 147ndash165 1976

[11] D Chinea ldquoAlmost contact metric submersionsrdquo Rendiconti delCircolo Matematico di Palermo II vol 34 no 1 pp 319ndash3301984

[12] D Chinea ldquoTransference of structures on almost complexcontact metric submersionsrdquo Houston Journal of Mathematicsvol 14 no 1 pp 9ndash22 1988

[13] Y Gunduzalp and B Sahin ldquoPara-contact para-complexpseudo-Riemannian submersionsrdquo Bulletin of the MalaysianMathematical Sciences Society

[14] I Sato ldquoOn a structure similar to the almost contact structurerdquoTensor vol 30 no 3 pp 219ndash224 1976

[15] P K Rashevskij ldquoThe scalar field in a stratified spacerdquoTrudy Seminara po Vektornomu i Tenzornomu Analizu s ikhPrilozheniyami k Geometrii Mekhanike i Fizike vol 6 pp 225ndash248 1948

[16] P Libermann ldquoSur les structures presque paracomplexesrdquoComptes Rendus de lrsquoAcademie des Sciences I vol 234 pp 2517ndash2519 1952

[17] E M Patterson ldquoRiemann extensions which have Kahlermetricsrdquo Proceedings of the Royal Society of Edinburgh AMathematics vol 64 pp 113ndash126 1954

[18] S Sasaki ldquoOn differentiable manifolds with certain structureswhich are closely related to almost contact structure Irdquo TheTohoku Mathematical Journal vol 12 pp 459ndash476 1960

[19] S Zamkovoy ldquoCanonical connections on paracontact mani-foldsrdquo Annals of Global Analysis and Geometry vol 36 no 1pp 37ndash60 2009

[20] D E Blair Riemannian Geometry of Contact and SymplecticManifolds vol 23 of Progress in Mathematics BirkhauserBoston Mass USA 2002

[21] V Cruceanu P Fortuny and P M Gadea ldquoA survey onparacomplex geometryrdquoThe Rocky Mountain Journal of Mathe-matics vol 26 no 1 pp 83ndash115 1996

[22] P M Gadea and J M Masque ldquoClassification of almost para-Hermitian manifoldsrdquo Rendiconti di Matematica e delle sueApplicazioni VII vol 7 no 11 pp 377ndash396 1991

[23] M Falcitelli S Ianus and A M Pastore Riemannian Submer-sions and Related Topics World Scientific River Edge NJ USA2004

[24] E G Rio and D N Kupeli Semi-Riemannian Maps andTheir Applications Kluwer Academic Publisher DordrechtTheNetherlands 1999

[25] B Sahin ldquoSemi-invariant submersions from almost HermitianmanifoldsrdquoCanadianMathematical Bulletin vol 54 no 3 2011

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 9: Research Article Paracomplex Paracontact Pseudo-Riemannian …downloads.hindawi.com/archive/2014/616487.pdf · 2019-07-31 · Research Article Paracomplex Paracontact Pseudo-Riemannian

Geometry 9

Theorem 17 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion and let the fibres of 119891 bepseudo-Riemannian submanifolds of 119872 Let 119883 119884 be basicvector fields 119891-related to 119883 119884 respectively If the total spaceis para-Kahler then the base space is paracosymplectic Theconverse is true if nabla

119883119869 is vertical

Proof We have for any 119883119884 isin Γ(119879119872) (nabla119883120601)119884 = 0 which

gives 119892(119885 (nabla119883120601)119884) = 0 for any 119885 isin Γ(119879119872)

From Proposition 14 we have

119892 (119885 (nabla119883120601)119884) + 120576120578 (119884) (nabla

119883120578)119885 + 120576120578 (119885) (nabla

119883120578) 119884 ∘ 119891

= 119892 (119885 (nabla119883119869) 119884)

(84)

Let nabla 119869 = 0 that is the total space is para-KahlerThen from(84) we obtain nabla120601 = 0 and nabla120578 = 0 Hence the base space isparacosymplectic

Again let (nabla119883120601)119884 = 0 and nabla120578 = 0 Then 119892(119885 (nabla

119883119869)119884) =

0 which implies that (nabla119883119869)119884 is a vertical vector field

Theorem 18 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion and let the fibres of 119891 bepseudo-Riemannian submanifolds of 119872 Let 119883 119884 and 119885 bebasic vector fields 119891-related to 119883 119884 and 119885 respectively If120578(119883)119885(120578(119884))+120578(119884)119883(120578(119885))minus120578(119883)119884(120578(119885)) = 0 then the totalspace is almost para-Kahler if and only if the base space119872 isan almost paracosymplectic manifold

Proof We have the following equation

3119889119865 (119883 119884 119885)

= 3 (119891lowast

lowast119889Φ) (119883 119884 119885) + 2120576120578 (119885) 119889120578 (119883 119884)

minus 2120576120578 (119884) 119889120578 (119883 119885)

+ 2120576120578 (119883) 119889120578 (119884 119885) + 2120576120578 (119883)119885 (120578 (119884))

+ 2120576120578 (119884)119883 (120578 (119885))

minus 2120576120578 (119883)119884 (120578 (119885))

(85)

If 119889120578 = 0 119889Φ = 0 and 120578(119883)119885(120578(119884)) + 120578(119884)119883(120578(119885)) minus

120578(119883)119884(120578(119885)) = 0 then from (85) we have 119889119865 = 0 Hencethe total space is almost para-Kahler

Conversely let 119889119865 = 0 and 120578(119883)119885(120578(119884))+120578(119884)119883(120578(119885))minus120578(119883)119884(120578(119885)) = 0

By using the above equation in (85) we have 119889120578 = 0 and119889Φ = 0

Hence the base space is almost paracosymplectic

Nowwe investigate the properties of fundamental tensorsT andA of a pseudo-Riemannian submersion

Lemma 19 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometric manifold119872 and

let the fibres of 119891 be pseudo-Riemannian submanifolds of 119872Then for any horizontal vector fields119883 119884 and for any verticalvector fields 119880 119881 on119872 one has

(i) A119883(119869 119884) = 119869(A

119883119884)

(ii) A119869119883(119884) = 119869(A

119883119884)

(iii) T119880(119869 119881) = 119869(T

119880119881)

(iv) T119869119880119881 = 119869(T

119880119881)

Proof The proof follows using similar steps as in Lemmas 3and 4 of [13] so we omit it

Lemma 20 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometric manifold119872 andlet the fibres of 119891 be pseudo-Riemannian submanifolds of 119872Then for any vector fields 119864 119865 on119872 one has

(i) A119864(119869119865) = 119869(A

119864119865)

(ii) T119864(119869119865) = 119869(T

119864119865)

Proof The proof follows from (37) and (38)

Theorem 21 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometric manifold119872 andlet the fibres of 119891 be pseudo-Riemannian submanifolds of 119872Then the horizontal distribution is integrable

Proof For any vertical vector field 119880 we have

119892 (119869 (A119883119884) 119880) = 119892 (A

119883119869 119884119880)

= minus119892 (119869119884A119883119880)

= minus119892 (119869119884 ℎ (nabla119880119883))

= 119892 (119884 ℎ (119869 (nabla119880119883)))

= 119892 (119884 ℎ (minusnabla119880119869)119883

+nabla119880(119869119883))

= 119892 (119884 ℎ nabla119880(119869119883)) = 119892 (119884A

119869119883119880)

= minus119892 (A119869119883119884119880) = minus119892 (119869 (A

119883119884) 119880)

(86)

Thus 119892(119869(A119883119884) 119880) = 0 which is true for all119883 and 119884

SoA119883119884 = 0

Hence the horizontal distribution is integrable

Theorem 22 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometric manifold119872 andlet the fibres of 119891 be pseudo-Riemannian submanifolds of 119872Then the submersion is an affine map onH

10 Geometry

Proof The second fundamental form of 119891 is defined by

(nabla119891lowast) (119864 119865) = (nabla

119891

119864119891lowast(119865)) ∘ 119891 minus 119891

lowast(nabla119864119865) (87)

where 119864 119865 isin Γ(119879119872) and nabla119891 is pullback connection of Levi-Civita connection nabla on119872 with respect to 119891

We have for any119883119884 isinH

(nabla119891lowast) (119883 119884) = (nabla

119891

119883

119891lowast(119884)) ∘ 119891 minus 119891

lowast(nabla119883119884) (88)

By using Lemma 1 we have 119891lowast(ℎ(nabla119883119884)) = (nabla

119883119884) ∘ 119891 which

implies nabla119891lowast= 0

Hence the submersion 119891 is an affine map onH

Theorem 23 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Hermitian man-ifold119872 onto an almost paracontact pseudometric manifold119872and let the fibres of 119891 be pseudo-Riemannian submanifolds of119872 Then the submersion is an affine map on V if and only ifthe fibres of 119891 are totally geodesic

Proof We have for any 119880119881 isinV

(nabla119891lowast) (119880 119881) = minus119891

lowast(ℎ (nabla119880119881)) (89)

which in view of (27) gives

(nabla119891lowast) (119880 119881) = minus119891

lowast(T119880119881) (90)

Let the fibres of 119891 be totally geodesic Then T = 0Consequently from the above equation we have nabla119891

lowast= 0

Thus the map 119891 is affine onVConversely let the submersion 119891 be an affine map onV

Then nabla119891lowast= 0 which impliesT = 0

Hence the fibres of 119891 are totally geodesic

Theorem 24 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Hermitian man-ifold119872 onto an almost paracontact pseudometric manifold119872and let the fibres of 119891 be pseudo-Riemannian submanifoldsof 119872 Then the submersion is an affine map if and only ifℎ(nabla119864ℎ119865)+A

ℎ119864V119865+TV119864V119865 is 119891-related to nabla

119883119884 for any 119864 119865 isin

Γ(119879119872)

Proof For any 119864 119865 isin Γ(119879119872) with 119891lowastℎ119864 = 119883 ∘ 119891 and 119891

lowastV119865 =

119884 ∘ 119891 we have

(nabla119891lowast) (119864 119865) = (nabla

119891lowastℎ119864(119891lowastℎ119865)) ∘ 119891 minus 119891

lowast(ℎ (nabla119864119865))

= (nabla119883119884) ∘ 119891 minus 119891

lowast(ℎ (nablaℎ119864ℎ119865 + nabla

ℎ119864V119865

+nablaV119864ℎ119865 + nablaV119864ℎ119865))

(91)

By using (27) and (31) in the above equation we have

(nabla119891lowast) (119864 119865) = (nabla

119883119884) ∘ 119891 minus 119891

lowast(ℎ (nabla119864ℎ119865) +A

ℎ119864V119865

+TV119864V119865) (92)

Let the submersion map be affine Then for any 119864 119865 isin

Γ(119879119872) (nabla119891lowast)(119864 119865) = 0 Equation (92) implies (nabla

119883119884) ∘ 119891 =

119891lowast(ℎ(nabla119864ℎ119865) +A

ℎ119864V119865 +TV119864ℎ119865)

Conversely let ℎ(nabla119864119865) + A

ℎ119864V119865 + TV119864ℎ119865 be 119891-related

to nabla119883119884 for any 119864 119865 isin Γ(119879119872) Then from (92) we have

(nabla119891lowast)(119864 119865) = 0Hence the submersion map 119891 is affine

4 Curvature Properties

In this section the paraholomorphic bisectional curvaturesand paraholomorphic sectional curvatures of total mani-fold base manifold and fibres of paracomplex paracontactpseudo-Riemannian submersion and their curvature proper-ties are studied

Let 119891 119872 rarr 119872 be a paracomplex paracontact pseudo-Riemannian submersion from an almost para-Hermitianmanifold (119872 119869 119892) onto an almost paracontact pseudometricmanifold (119872 120601 120585 120578 119892)

Suppose that the vector fields 119864 119865 span the 2-dimensional plane at point 119901 of 119872 and let R be theRiemannian curvature tensor of 119872 The paraholomorphicbisectional curvature 119861(119864 119865) of 119872 for any pair of nonzeronon-lightlike vector fields 119864 119865 on 119872 is defined by theformula

119861 (119864 119865) =

R (119864 119869119864 119865 119869119865)

119892 (119864 119864) 119892 (119865 119865) (93)

For a nonzero non-lightlike vector field 119864 the vector field119869119864 is also non-lightlike and 119864 119869119864 span the 2-dimensionalplane Then the paraholomorphic sectional curvature 119867(119864)is defined as

119867(119864) = 119861 (119864 119864) =

R (119864 119869119864 119864 119869119864)

119892 (119864 119864) 119892 (119864 119864) (94)

The curvature properties of Riemannian submersion andsemi-Riemannian submersion have been extensively studiedin the work of OrsquoNeill [1] and Gray [3]

Let 119861ℎ and 119861V be the paraholomorphic bisectional cur-vatures of horizontal and vertical spaces respectively Let119867ℎ and 119867V be the paraholomorphic sectional curvatures of

horizontal and vertical spaces respectively Let 119861 and 119867 bethe paraholomorphic bisectional and sectional curvatures ofthe base manifold respectively

Proposition 25 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion from an almostpara-Hermitian manifold onto an almost paracontact pseudo-metric manifold and let the fibres of 119891 be pseudo-Riemanniansubmanifolds of 119872 Let 119880 119881 be non-lightlike unit verticalvector fields and let119883 119884 be non-lightlike unit horizontal vectorfields on119872 Then one has

119861 (119880119881) = 119861V(119880 119881) + 119892 (T

119880(119869119881) T

119869119880119881)

minus 119892 (T119869119880(119869119881) T

119880119881)

(95)

Geometry 11

119861 (119883119880) = 119892 ((nabla119880A)119883

119869119883 119869119880) minus 119892 (A119883119869119880A

119869119883119880)

+ 119892 (A119883119880A119869119883119869119880) minus 119892 ((nabla

119869119880A)119883

119869119883119880)

+ 119892 (T119869119880119883T119880(119869119883))

minus 119892 (T119880119883T119869119880(119869119883))

(96)

119861 (119883 119884) = 119861ℎ

(119883 119884) minus 2119892 (A119883(119869119883) A

119884(119869 119884))

+ 119892 (A119869119883119884A119883(119869 119884))

minus 119892 (A119883119884A119869119883(119869 119884))

(97)

Proof Using Definitions (93) and (94) of paraholomorphicsectional curvature and fundamental equations of submer-sion obtained by OrsquoNeill [1] we have (95) (96) and (97)

Corollary 26 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion from an almostpara-Hermitian manifold onto an almost paracontact pseudo-metric manifold If the fibres of 119891 are totally geodesic pseudo-Riemannian submanifolds of119872 then for any non-lightlike unitvertical vector fields 119880 and 119881 one has

119861 (119880119881) = 119861V(119880 119881) (98)

Corollary 27 Let 119891 119872 rarr 119872 be a paracomplex para-contact pseudo-Riemannian submersion from an almost para-Hermitian manifold onto an almost paracontact pseudometricmanifold and let the fibres of 119891 be totally geodesic pseudo-Riemannian submanifolds of119872 If the horizontal distributionis integrable then for any non-lightlike unit horizontal vectorfields119883 and 119884 one has

119861 (119883 119884) = 119861ℎ

(119883 119884) (99)

Proposition 28 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion from an almostpara-Hermitian manifold onto an almost paracontact pseudo-metric manifold and let the fibres of 119891 be pseudo-Riemanniansubmanifolds of119872 Let 119880 and 119883 be non-lightlike unit verticalvector field and non-lightlike unit horizontal vector fieldrespectively Then one has

119867(119880) = 119867V(119880) +

10038171003817100381710038171003817T119880(119869119880)

10038171003817100381710038171003817

2

minus 119892 (T119869119880(119869119880) T

119880119880)

(100)

119867(119883) = 119867 (119883) ∘ 119891 minus 310038171003817100381710038171003817A119883(119869119883)

10038171003817100381710038171003817

2

(101)

Proof The proof is straightforward If we take 119880 = 119881 in (95)and119883 = 119884 in (97) we have (98) and (99)

Corollary 29 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion from an almostpara-Hermitian manifold onto an almost paracontact pseudo-metric manifold If the fibres of 119891 are totally geodesic pseudo-Riemannian submanifolds of 119872 then the total manifold andfibres of119891have the same paraholomorphic sectional curvatures

Proof Since the fibres are totally geodesic T = 0 conse-quently we have

119867(119880) = 119867V(119880) (102)

Corollary 30 Let 119891 119872 rarr 119872 be a paracomplex para-contact pseudo-Riemannian submersion from an almost para-Hermitian manifold onto an almost paracontact pseudometricmanifold and let the fibres of 119891 be totally geodesic pseudo-Riemannian submanifolds of119872 If the horizontal distributionis integrable then the base manifold and horizontal distribu-tion have the same paraholomorphic sectional curvatures

Proof Since the horizontal distribution is integrableA = 0consequently we have

119867(119883) = 119867 (119883) ∘ 119891 (103)

Theorem 31 Let 119891 119872119898

rarr 119872119899 be a paracomplex

paracontact pseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometricmanifold 119872 and let the fibres of 119891 be pseudo-Riemanniansubmanifolds of 119872 If 119880 119881 are the non-lightlike unit verticalvector fields and 119883 119884 are the non-lightlike unit horizontalvector fields then one has

119861 (119880119881) = 119861V(119880 119881) (104)

119861 (119883119880) = minus210038171003817100381710038171003817T11988011988310038171003817100381710038171003817

2

(105)

119861 (119883 119884) = 119861 (119883 119884) ∘ 119891 (106)

Proof Using results of Lemma 19 in (95) we have

119861 (119880119881) = 119861V(119880 119881) minus 119892 (119869 (T

119880119881) 119869 (T

119880119881))

minus 119892 (1198692

(T119880119881) T

119880119881)

= 119861 (119880119881) + 119892 (T119880119881T119880119881) minus 119892 (T

119880119881T119880119881)

= 119861V(119880 119881)

(107)

Applying results of Lemma 19 in (96) we have

119861 (119883119880) = 119892 ((nabla119880A)119883

(119869119883) 119869119880) minus 119892 ((nabla119869119880A)119883

(119869119883) 119880)

+ 21003817100381710038171003817A119883119880

1003817100381710038171003817

2

minus 210038171003817100381710038171003817T11988011988310038171003817100381710038171003817

2

(108)

Since byTheorem 21 the horizontal distribution is integrablewe haveA = 0 which implies

119861 (119883119880) = minus210038171003817100381710038171003817T11988011988310038171003817100381710038171003817

2

(109)

In view ofA = 0 (104) follows from (97)

12 Geometry

Theorem 32 Let 119891 119872119898

rarr 119872119899 be a paracomplex

paracontact pseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometricmanifold 119872 and let the fibres of 119891 be pseudo-Riemanniansubmanifolds of119872 If 119880 119883 are non-lightlike unit vertical andnon-lightlike unit horizontal vector fields respectively then onehas

119867(119880) = 119867V(119880) minus 2

1003817100381710038171003817T1198801198801003817100381710038171003817

2

(110)

119867(119883) = 119867 (119883) ∘ 119891 (111)

Proof Since 119891 is the paracomplex paracontact pseudo-Riemannian submersion from a para-Kahler manifold 119872

onto an almost paracontact pseudometric manifold 119872 by(16) and equations of Lemma 19 we have

119892 (T119869119880(119869119880) T

119880119880) = 119892 (119869

2

(T119880119880) T

119880119880) =

1003817100381710038171003817T1198801198801003817100381710038171003817

2

119892 (T119880(119869119880) T

119880(119869119880)) = minus119892 (T

119880119880T119880119880) = minus

1003817100381710038171003817T1198801198801003817100381710038171003817

2

(112)

and by using the above results in (100) we have

119867(119880) = 119867V(119880) minus

1003817100381710038171003817T1198801198801003817100381710038171003817

2

minus1003817100381710038171003817T119880119880

1003817100381710038171003817

2

= 119867V(119880) minus 2

1003817100381710038171003817T1198801198801003817100381710038171003817

2

(113)

Again since horizontal distribution is integrable we haveA = 0 and putting it in (101) we obtain (111)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Uma Shankar Verma is thankful to University Grant Com-mission New Delhi India for financial support

References

[1] B OrsquoNeill ldquoThe fundamental equations of a submersionrdquo TheMichigan Mathematical Journal vol 13 pp 459ndash469 1966

[2] B OrsquoNeill Semi-Riemannian Geometry with Applications toRelativity vol 103 of Pure and Applied Mathematics AcademicPress New York NY USA 1983

[3] A Gray ldquoPseudo-Riemannian almost product manifolds andsubmersionsrdquo vol 16 pp 715ndash737 1967

[4] J P Bourguignon and H B Lawson A Mathematicians Visitto Kaluza-Klein Theory Rendiconti del Seminario Matematico1989

[5] S Ianus and M Visinescu ldquoSpace-time compactification andRiemannian submersionsrdquo in The Mathematical Heritage GRassias and C F Gauss Eds pp 358ndash371 World ScientificRiver Edge NJ USA 1991

[6] J P Bourguignon and H B Lawson Jr ldquoStability and isolationphenomena for Yang-Mills fieldsrdquo Communications in Mathe-matical Physics vol 79 no 2 pp 189ndash230 1981

[7] B Watson ldquoG 1198661015840

-Riemannian submersions and nonlineargauge field equations of general relativityrdquo in Global AnalysismdashAnalysis on Manifolds T Rassias and M Morse Eds vol 57 ofTeubner-Texte zur Mathematik pp 324ndash349 Teubner LeipzigGermany 1983

[8] C Altafini ldquoRedundant robotic chains on Riemannian submer-sionsrdquo IEEE Transactions on Robotics and Automation vol 20no 2 pp 335ndash340 2004

[9] M T Mustafa ldquoApplications of harmonic morphisms to grav-ityrdquo Journal of Mathematical Physics vol 41 no 10 pp 6918ndash6929 2000

[10] B Watson ldquoAlmost Hermitian submersionsrdquo Journal of Differ-ential Geometry vol 11 no 1 pp 147ndash165 1976

[11] D Chinea ldquoAlmost contact metric submersionsrdquo Rendiconti delCircolo Matematico di Palermo II vol 34 no 1 pp 319ndash3301984

[12] D Chinea ldquoTransference of structures on almost complexcontact metric submersionsrdquo Houston Journal of Mathematicsvol 14 no 1 pp 9ndash22 1988

[13] Y Gunduzalp and B Sahin ldquoPara-contact para-complexpseudo-Riemannian submersionsrdquo Bulletin of the MalaysianMathematical Sciences Society

[14] I Sato ldquoOn a structure similar to the almost contact structurerdquoTensor vol 30 no 3 pp 219ndash224 1976

[15] P K Rashevskij ldquoThe scalar field in a stratified spacerdquoTrudy Seminara po Vektornomu i Tenzornomu Analizu s ikhPrilozheniyami k Geometrii Mekhanike i Fizike vol 6 pp 225ndash248 1948

[16] P Libermann ldquoSur les structures presque paracomplexesrdquoComptes Rendus de lrsquoAcademie des Sciences I vol 234 pp 2517ndash2519 1952

[17] E M Patterson ldquoRiemann extensions which have Kahlermetricsrdquo Proceedings of the Royal Society of Edinburgh AMathematics vol 64 pp 113ndash126 1954

[18] S Sasaki ldquoOn differentiable manifolds with certain structureswhich are closely related to almost contact structure Irdquo TheTohoku Mathematical Journal vol 12 pp 459ndash476 1960

[19] S Zamkovoy ldquoCanonical connections on paracontact mani-foldsrdquo Annals of Global Analysis and Geometry vol 36 no 1pp 37ndash60 2009

[20] D E Blair Riemannian Geometry of Contact and SymplecticManifolds vol 23 of Progress in Mathematics BirkhauserBoston Mass USA 2002

[21] V Cruceanu P Fortuny and P M Gadea ldquoA survey onparacomplex geometryrdquoThe Rocky Mountain Journal of Mathe-matics vol 26 no 1 pp 83ndash115 1996

[22] P M Gadea and J M Masque ldquoClassification of almost para-Hermitian manifoldsrdquo Rendiconti di Matematica e delle sueApplicazioni VII vol 7 no 11 pp 377ndash396 1991

[23] M Falcitelli S Ianus and A M Pastore Riemannian Submer-sions and Related Topics World Scientific River Edge NJ USA2004

[24] E G Rio and D N Kupeli Semi-Riemannian Maps andTheir Applications Kluwer Academic Publisher DordrechtTheNetherlands 1999

[25] B Sahin ldquoSemi-invariant submersions from almost HermitianmanifoldsrdquoCanadianMathematical Bulletin vol 54 no 3 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 10: Research Article Paracomplex Paracontact Pseudo-Riemannian …downloads.hindawi.com/archive/2014/616487.pdf · 2019-07-31 · Research Article Paracomplex Paracontact Pseudo-Riemannian

10 Geometry

Proof The second fundamental form of 119891 is defined by

(nabla119891lowast) (119864 119865) = (nabla

119891

119864119891lowast(119865)) ∘ 119891 minus 119891

lowast(nabla119864119865) (87)

where 119864 119865 isin Γ(119879119872) and nabla119891 is pullback connection of Levi-Civita connection nabla on119872 with respect to 119891

We have for any119883119884 isinH

(nabla119891lowast) (119883 119884) = (nabla

119891

119883

119891lowast(119884)) ∘ 119891 minus 119891

lowast(nabla119883119884) (88)

By using Lemma 1 we have 119891lowast(ℎ(nabla119883119884)) = (nabla

119883119884) ∘ 119891 which

implies nabla119891lowast= 0

Hence the submersion 119891 is an affine map onH

Theorem 23 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Hermitian man-ifold119872 onto an almost paracontact pseudometric manifold119872and let the fibres of 119891 be pseudo-Riemannian submanifolds of119872 Then the submersion is an affine map on V if and only ifthe fibres of 119891 are totally geodesic

Proof We have for any 119880119881 isinV

(nabla119891lowast) (119880 119881) = minus119891

lowast(ℎ (nabla119880119881)) (89)

which in view of (27) gives

(nabla119891lowast) (119880 119881) = minus119891

lowast(T119880119881) (90)

Let the fibres of 119891 be totally geodesic Then T = 0Consequently from the above equation we have nabla119891

lowast= 0

Thus the map 119891 is affine onVConversely let the submersion 119891 be an affine map onV

Then nabla119891lowast= 0 which impliesT = 0

Hence the fibres of 119891 are totally geodesic

Theorem 24 Let 119891 119872 rarr 119872 be a paracomplex paracontactpseudo-Riemannian submersion from a para-Hermitian man-ifold119872 onto an almost paracontact pseudometric manifold119872and let the fibres of 119891 be pseudo-Riemannian submanifoldsof 119872 Then the submersion is an affine map if and only ifℎ(nabla119864ℎ119865)+A

ℎ119864V119865+TV119864V119865 is 119891-related to nabla

119883119884 for any 119864 119865 isin

Γ(119879119872)

Proof For any 119864 119865 isin Γ(119879119872) with 119891lowastℎ119864 = 119883 ∘ 119891 and 119891

lowastV119865 =

119884 ∘ 119891 we have

(nabla119891lowast) (119864 119865) = (nabla

119891lowastℎ119864(119891lowastℎ119865)) ∘ 119891 minus 119891

lowast(ℎ (nabla119864119865))

= (nabla119883119884) ∘ 119891 minus 119891

lowast(ℎ (nablaℎ119864ℎ119865 + nabla

ℎ119864V119865

+nablaV119864ℎ119865 + nablaV119864ℎ119865))

(91)

By using (27) and (31) in the above equation we have

(nabla119891lowast) (119864 119865) = (nabla

119883119884) ∘ 119891 minus 119891

lowast(ℎ (nabla119864ℎ119865) +A

ℎ119864V119865

+TV119864V119865) (92)

Let the submersion map be affine Then for any 119864 119865 isin

Γ(119879119872) (nabla119891lowast)(119864 119865) = 0 Equation (92) implies (nabla

119883119884) ∘ 119891 =

119891lowast(ℎ(nabla119864ℎ119865) +A

ℎ119864V119865 +TV119864ℎ119865)

Conversely let ℎ(nabla119864119865) + A

ℎ119864V119865 + TV119864ℎ119865 be 119891-related

to nabla119883119884 for any 119864 119865 isin Γ(119879119872) Then from (92) we have

(nabla119891lowast)(119864 119865) = 0Hence the submersion map 119891 is affine

4 Curvature Properties

In this section the paraholomorphic bisectional curvaturesand paraholomorphic sectional curvatures of total mani-fold base manifold and fibres of paracomplex paracontactpseudo-Riemannian submersion and their curvature proper-ties are studied

Let 119891 119872 rarr 119872 be a paracomplex paracontact pseudo-Riemannian submersion from an almost para-Hermitianmanifold (119872 119869 119892) onto an almost paracontact pseudometricmanifold (119872 120601 120585 120578 119892)

Suppose that the vector fields 119864 119865 span the 2-dimensional plane at point 119901 of 119872 and let R be theRiemannian curvature tensor of 119872 The paraholomorphicbisectional curvature 119861(119864 119865) of 119872 for any pair of nonzeronon-lightlike vector fields 119864 119865 on 119872 is defined by theformula

119861 (119864 119865) =

R (119864 119869119864 119865 119869119865)

119892 (119864 119864) 119892 (119865 119865) (93)

For a nonzero non-lightlike vector field 119864 the vector field119869119864 is also non-lightlike and 119864 119869119864 span the 2-dimensionalplane Then the paraholomorphic sectional curvature 119867(119864)is defined as

119867(119864) = 119861 (119864 119864) =

R (119864 119869119864 119864 119869119864)

119892 (119864 119864) 119892 (119864 119864) (94)

The curvature properties of Riemannian submersion andsemi-Riemannian submersion have been extensively studiedin the work of OrsquoNeill [1] and Gray [3]

Let 119861ℎ and 119861V be the paraholomorphic bisectional cur-vatures of horizontal and vertical spaces respectively Let119867ℎ and 119867V be the paraholomorphic sectional curvatures of

horizontal and vertical spaces respectively Let 119861 and 119867 bethe paraholomorphic bisectional and sectional curvatures ofthe base manifold respectively

Proposition 25 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion from an almostpara-Hermitian manifold onto an almost paracontact pseudo-metric manifold and let the fibres of 119891 be pseudo-Riemanniansubmanifolds of 119872 Let 119880 119881 be non-lightlike unit verticalvector fields and let119883 119884 be non-lightlike unit horizontal vectorfields on119872 Then one has

119861 (119880119881) = 119861V(119880 119881) + 119892 (T

119880(119869119881) T

119869119880119881)

minus 119892 (T119869119880(119869119881) T

119880119881)

(95)

Geometry 11

119861 (119883119880) = 119892 ((nabla119880A)119883

119869119883 119869119880) minus 119892 (A119883119869119880A

119869119883119880)

+ 119892 (A119883119880A119869119883119869119880) minus 119892 ((nabla

119869119880A)119883

119869119883119880)

+ 119892 (T119869119880119883T119880(119869119883))

minus 119892 (T119880119883T119869119880(119869119883))

(96)

119861 (119883 119884) = 119861ℎ

(119883 119884) minus 2119892 (A119883(119869119883) A

119884(119869 119884))

+ 119892 (A119869119883119884A119883(119869 119884))

minus 119892 (A119883119884A119869119883(119869 119884))

(97)

Proof Using Definitions (93) and (94) of paraholomorphicsectional curvature and fundamental equations of submer-sion obtained by OrsquoNeill [1] we have (95) (96) and (97)

Corollary 26 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion from an almostpara-Hermitian manifold onto an almost paracontact pseudo-metric manifold If the fibres of 119891 are totally geodesic pseudo-Riemannian submanifolds of119872 then for any non-lightlike unitvertical vector fields 119880 and 119881 one has

119861 (119880119881) = 119861V(119880 119881) (98)

Corollary 27 Let 119891 119872 rarr 119872 be a paracomplex para-contact pseudo-Riemannian submersion from an almost para-Hermitian manifold onto an almost paracontact pseudometricmanifold and let the fibres of 119891 be totally geodesic pseudo-Riemannian submanifolds of119872 If the horizontal distributionis integrable then for any non-lightlike unit horizontal vectorfields119883 and 119884 one has

119861 (119883 119884) = 119861ℎ

(119883 119884) (99)

Proposition 28 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion from an almostpara-Hermitian manifold onto an almost paracontact pseudo-metric manifold and let the fibres of 119891 be pseudo-Riemanniansubmanifolds of119872 Let 119880 and 119883 be non-lightlike unit verticalvector field and non-lightlike unit horizontal vector fieldrespectively Then one has

119867(119880) = 119867V(119880) +

10038171003817100381710038171003817T119880(119869119880)

10038171003817100381710038171003817

2

minus 119892 (T119869119880(119869119880) T

119880119880)

(100)

119867(119883) = 119867 (119883) ∘ 119891 minus 310038171003817100381710038171003817A119883(119869119883)

10038171003817100381710038171003817

2

(101)

Proof The proof is straightforward If we take 119880 = 119881 in (95)and119883 = 119884 in (97) we have (98) and (99)

Corollary 29 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion from an almostpara-Hermitian manifold onto an almost paracontact pseudo-metric manifold If the fibres of 119891 are totally geodesic pseudo-Riemannian submanifolds of 119872 then the total manifold andfibres of119891have the same paraholomorphic sectional curvatures

Proof Since the fibres are totally geodesic T = 0 conse-quently we have

119867(119880) = 119867V(119880) (102)

Corollary 30 Let 119891 119872 rarr 119872 be a paracomplex para-contact pseudo-Riemannian submersion from an almost para-Hermitian manifold onto an almost paracontact pseudometricmanifold and let the fibres of 119891 be totally geodesic pseudo-Riemannian submanifolds of119872 If the horizontal distributionis integrable then the base manifold and horizontal distribu-tion have the same paraholomorphic sectional curvatures

Proof Since the horizontal distribution is integrableA = 0consequently we have

119867(119883) = 119867 (119883) ∘ 119891 (103)

Theorem 31 Let 119891 119872119898

rarr 119872119899 be a paracomplex

paracontact pseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometricmanifold 119872 and let the fibres of 119891 be pseudo-Riemanniansubmanifolds of 119872 If 119880 119881 are the non-lightlike unit verticalvector fields and 119883 119884 are the non-lightlike unit horizontalvector fields then one has

119861 (119880119881) = 119861V(119880 119881) (104)

119861 (119883119880) = minus210038171003817100381710038171003817T11988011988310038171003817100381710038171003817

2

(105)

119861 (119883 119884) = 119861 (119883 119884) ∘ 119891 (106)

Proof Using results of Lemma 19 in (95) we have

119861 (119880119881) = 119861V(119880 119881) minus 119892 (119869 (T

119880119881) 119869 (T

119880119881))

minus 119892 (1198692

(T119880119881) T

119880119881)

= 119861 (119880119881) + 119892 (T119880119881T119880119881) minus 119892 (T

119880119881T119880119881)

= 119861V(119880 119881)

(107)

Applying results of Lemma 19 in (96) we have

119861 (119883119880) = 119892 ((nabla119880A)119883

(119869119883) 119869119880) minus 119892 ((nabla119869119880A)119883

(119869119883) 119880)

+ 21003817100381710038171003817A119883119880

1003817100381710038171003817

2

minus 210038171003817100381710038171003817T11988011988310038171003817100381710038171003817

2

(108)

Since byTheorem 21 the horizontal distribution is integrablewe haveA = 0 which implies

119861 (119883119880) = minus210038171003817100381710038171003817T11988011988310038171003817100381710038171003817

2

(109)

In view ofA = 0 (104) follows from (97)

12 Geometry

Theorem 32 Let 119891 119872119898

rarr 119872119899 be a paracomplex

paracontact pseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometricmanifold 119872 and let the fibres of 119891 be pseudo-Riemanniansubmanifolds of119872 If 119880 119883 are non-lightlike unit vertical andnon-lightlike unit horizontal vector fields respectively then onehas

119867(119880) = 119867V(119880) minus 2

1003817100381710038171003817T1198801198801003817100381710038171003817

2

(110)

119867(119883) = 119867 (119883) ∘ 119891 (111)

Proof Since 119891 is the paracomplex paracontact pseudo-Riemannian submersion from a para-Kahler manifold 119872

onto an almost paracontact pseudometric manifold 119872 by(16) and equations of Lemma 19 we have

119892 (T119869119880(119869119880) T

119880119880) = 119892 (119869

2

(T119880119880) T

119880119880) =

1003817100381710038171003817T1198801198801003817100381710038171003817

2

119892 (T119880(119869119880) T

119880(119869119880)) = minus119892 (T

119880119880T119880119880) = minus

1003817100381710038171003817T1198801198801003817100381710038171003817

2

(112)

and by using the above results in (100) we have

119867(119880) = 119867V(119880) minus

1003817100381710038171003817T1198801198801003817100381710038171003817

2

minus1003817100381710038171003817T119880119880

1003817100381710038171003817

2

= 119867V(119880) minus 2

1003817100381710038171003817T1198801198801003817100381710038171003817

2

(113)

Again since horizontal distribution is integrable we haveA = 0 and putting it in (101) we obtain (111)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Uma Shankar Verma is thankful to University Grant Com-mission New Delhi India for financial support

References

[1] B OrsquoNeill ldquoThe fundamental equations of a submersionrdquo TheMichigan Mathematical Journal vol 13 pp 459ndash469 1966

[2] B OrsquoNeill Semi-Riemannian Geometry with Applications toRelativity vol 103 of Pure and Applied Mathematics AcademicPress New York NY USA 1983

[3] A Gray ldquoPseudo-Riemannian almost product manifolds andsubmersionsrdquo vol 16 pp 715ndash737 1967

[4] J P Bourguignon and H B Lawson A Mathematicians Visitto Kaluza-Klein Theory Rendiconti del Seminario Matematico1989

[5] S Ianus and M Visinescu ldquoSpace-time compactification andRiemannian submersionsrdquo in The Mathematical Heritage GRassias and C F Gauss Eds pp 358ndash371 World ScientificRiver Edge NJ USA 1991

[6] J P Bourguignon and H B Lawson Jr ldquoStability and isolationphenomena for Yang-Mills fieldsrdquo Communications in Mathe-matical Physics vol 79 no 2 pp 189ndash230 1981

[7] B Watson ldquoG 1198661015840

-Riemannian submersions and nonlineargauge field equations of general relativityrdquo in Global AnalysismdashAnalysis on Manifolds T Rassias and M Morse Eds vol 57 ofTeubner-Texte zur Mathematik pp 324ndash349 Teubner LeipzigGermany 1983

[8] C Altafini ldquoRedundant robotic chains on Riemannian submer-sionsrdquo IEEE Transactions on Robotics and Automation vol 20no 2 pp 335ndash340 2004

[9] M T Mustafa ldquoApplications of harmonic morphisms to grav-ityrdquo Journal of Mathematical Physics vol 41 no 10 pp 6918ndash6929 2000

[10] B Watson ldquoAlmost Hermitian submersionsrdquo Journal of Differ-ential Geometry vol 11 no 1 pp 147ndash165 1976

[11] D Chinea ldquoAlmost contact metric submersionsrdquo Rendiconti delCircolo Matematico di Palermo II vol 34 no 1 pp 319ndash3301984

[12] D Chinea ldquoTransference of structures on almost complexcontact metric submersionsrdquo Houston Journal of Mathematicsvol 14 no 1 pp 9ndash22 1988

[13] Y Gunduzalp and B Sahin ldquoPara-contact para-complexpseudo-Riemannian submersionsrdquo Bulletin of the MalaysianMathematical Sciences Society

[14] I Sato ldquoOn a structure similar to the almost contact structurerdquoTensor vol 30 no 3 pp 219ndash224 1976

[15] P K Rashevskij ldquoThe scalar field in a stratified spacerdquoTrudy Seminara po Vektornomu i Tenzornomu Analizu s ikhPrilozheniyami k Geometrii Mekhanike i Fizike vol 6 pp 225ndash248 1948

[16] P Libermann ldquoSur les structures presque paracomplexesrdquoComptes Rendus de lrsquoAcademie des Sciences I vol 234 pp 2517ndash2519 1952

[17] E M Patterson ldquoRiemann extensions which have Kahlermetricsrdquo Proceedings of the Royal Society of Edinburgh AMathematics vol 64 pp 113ndash126 1954

[18] S Sasaki ldquoOn differentiable manifolds with certain structureswhich are closely related to almost contact structure Irdquo TheTohoku Mathematical Journal vol 12 pp 459ndash476 1960

[19] S Zamkovoy ldquoCanonical connections on paracontact mani-foldsrdquo Annals of Global Analysis and Geometry vol 36 no 1pp 37ndash60 2009

[20] D E Blair Riemannian Geometry of Contact and SymplecticManifolds vol 23 of Progress in Mathematics BirkhauserBoston Mass USA 2002

[21] V Cruceanu P Fortuny and P M Gadea ldquoA survey onparacomplex geometryrdquoThe Rocky Mountain Journal of Mathe-matics vol 26 no 1 pp 83ndash115 1996

[22] P M Gadea and J M Masque ldquoClassification of almost para-Hermitian manifoldsrdquo Rendiconti di Matematica e delle sueApplicazioni VII vol 7 no 11 pp 377ndash396 1991

[23] M Falcitelli S Ianus and A M Pastore Riemannian Submer-sions and Related Topics World Scientific River Edge NJ USA2004

[24] E G Rio and D N Kupeli Semi-Riemannian Maps andTheir Applications Kluwer Academic Publisher DordrechtTheNetherlands 1999

[25] B Sahin ldquoSemi-invariant submersions from almost HermitianmanifoldsrdquoCanadianMathematical Bulletin vol 54 no 3 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 11: Research Article Paracomplex Paracontact Pseudo-Riemannian …downloads.hindawi.com/archive/2014/616487.pdf · 2019-07-31 · Research Article Paracomplex Paracontact Pseudo-Riemannian

Geometry 11

119861 (119883119880) = 119892 ((nabla119880A)119883

119869119883 119869119880) minus 119892 (A119883119869119880A

119869119883119880)

+ 119892 (A119883119880A119869119883119869119880) minus 119892 ((nabla

119869119880A)119883

119869119883119880)

+ 119892 (T119869119880119883T119880(119869119883))

minus 119892 (T119880119883T119869119880(119869119883))

(96)

119861 (119883 119884) = 119861ℎ

(119883 119884) minus 2119892 (A119883(119869119883) A

119884(119869 119884))

+ 119892 (A119869119883119884A119883(119869 119884))

minus 119892 (A119883119884A119869119883(119869 119884))

(97)

Proof Using Definitions (93) and (94) of paraholomorphicsectional curvature and fundamental equations of submer-sion obtained by OrsquoNeill [1] we have (95) (96) and (97)

Corollary 26 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion from an almostpara-Hermitian manifold onto an almost paracontact pseudo-metric manifold If the fibres of 119891 are totally geodesic pseudo-Riemannian submanifolds of119872 then for any non-lightlike unitvertical vector fields 119880 and 119881 one has

119861 (119880119881) = 119861V(119880 119881) (98)

Corollary 27 Let 119891 119872 rarr 119872 be a paracomplex para-contact pseudo-Riemannian submersion from an almost para-Hermitian manifold onto an almost paracontact pseudometricmanifold and let the fibres of 119891 be totally geodesic pseudo-Riemannian submanifolds of119872 If the horizontal distributionis integrable then for any non-lightlike unit horizontal vectorfields119883 and 119884 one has

119861 (119883 119884) = 119861ℎ

(119883 119884) (99)

Proposition 28 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion from an almostpara-Hermitian manifold onto an almost paracontact pseudo-metric manifold and let the fibres of 119891 be pseudo-Riemanniansubmanifolds of119872 Let 119880 and 119883 be non-lightlike unit verticalvector field and non-lightlike unit horizontal vector fieldrespectively Then one has

119867(119880) = 119867V(119880) +

10038171003817100381710038171003817T119880(119869119880)

10038171003817100381710038171003817

2

minus 119892 (T119869119880(119869119880) T

119880119880)

(100)

119867(119883) = 119867 (119883) ∘ 119891 minus 310038171003817100381710038171003817A119883(119869119883)

10038171003817100381710038171003817

2

(101)

Proof The proof is straightforward If we take 119880 = 119881 in (95)and119883 = 119884 in (97) we have (98) and (99)

Corollary 29 Let 119891 119872 rarr 119872 be a paracomplexparacontact pseudo-Riemannian submersion from an almostpara-Hermitian manifold onto an almost paracontact pseudo-metric manifold If the fibres of 119891 are totally geodesic pseudo-Riemannian submanifolds of 119872 then the total manifold andfibres of119891have the same paraholomorphic sectional curvatures

Proof Since the fibres are totally geodesic T = 0 conse-quently we have

119867(119880) = 119867V(119880) (102)

Corollary 30 Let 119891 119872 rarr 119872 be a paracomplex para-contact pseudo-Riemannian submersion from an almost para-Hermitian manifold onto an almost paracontact pseudometricmanifold and let the fibres of 119891 be totally geodesic pseudo-Riemannian submanifolds of119872 If the horizontal distributionis integrable then the base manifold and horizontal distribu-tion have the same paraholomorphic sectional curvatures

Proof Since the horizontal distribution is integrableA = 0consequently we have

119867(119883) = 119867 (119883) ∘ 119891 (103)

Theorem 31 Let 119891 119872119898

rarr 119872119899 be a paracomplex

paracontact pseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometricmanifold 119872 and let the fibres of 119891 be pseudo-Riemanniansubmanifolds of 119872 If 119880 119881 are the non-lightlike unit verticalvector fields and 119883 119884 are the non-lightlike unit horizontalvector fields then one has

119861 (119880119881) = 119861V(119880 119881) (104)

119861 (119883119880) = minus210038171003817100381710038171003817T11988011988310038171003817100381710038171003817

2

(105)

119861 (119883 119884) = 119861 (119883 119884) ∘ 119891 (106)

Proof Using results of Lemma 19 in (95) we have

119861 (119880119881) = 119861V(119880 119881) minus 119892 (119869 (T

119880119881) 119869 (T

119880119881))

minus 119892 (1198692

(T119880119881) T

119880119881)

= 119861 (119880119881) + 119892 (T119880119881T119880119881) minus 119892 (T

119880119881T119880119881)

= 119861V(119880 119881)

(107)

Applying results of Lemma 19 in (96) we have

119861 (119883119880) = 119892 ((nabla119880A)119883

(119869119883) 119869119880) minus 119892 ((nabla119869119880A)119883

(119869119883) 119880)

+ 21003817100381710038171003817A119883119880

1003817100381710038171003817

2

minus 210038171003817100381710038171003817T11988011988310038171003817100381710038171003817

2

(108)

Since byTheorem 21 the horizontal distribution is integrablewe haveA = 0 which implies

119861 (119883119880) = minus210038171003817100381710038171003817T11988011988310038171003817100381710038171003817

2

(109)

In view ofA = 0 (104) follows from (97)

12 Geometry

Theorem 32 Let 119891 119872119898

rarr 119872119899 be a paracomplex

paracontact pseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometricmanifold 119872 and let the fibres of 119891 be pseudo-Riemanniansubmanifolds of119872 If 119880 119883 are non-lightlike unit vertical andnon-lightlike unit horizontal vector fields respectively then onehas

119867(119880) = 119867V(119880) minus 2

1003817100381710038171003817T1198801198801003817100381710038171003817

2

(110)

119867(119883) = 119867 (119883) ∘ 119891 (111)

Proof Since 119891 is the paracomplex paracontact pseudo-Riemannian submersion from a para-Kahler manifold 119872

onto an almost paracontact pseudometric manifold 119872 by(16) and equations of Lemma 19 we have

119892 (T119869119880(119869119880) T

119880119880) = 119892 (119869

2

(T119880119880) T

119880119880) =

1003817100381710038171003817T1198801198801003817100381710038171003817

2

119892 (T119880(119869119880) T

119880(119869119880)) = minus119892 (T

119880119880T119880119880) = minus

1003817100381710038171003817T1198801198801003817100381710038171003817

2

(112)

and by using the above results in (100) we have

119867(119880) = 119867V(119880) minus

1003817100381710038171003817T1198801198801003817100381710038171003817

2

minus1003817100381710038171003817T119880119880

1003817100381710038171003817

2

= 119867V(119880) minus 2

1003817100381710038171003817T1198801198801003817100381710038171003817

2

(113)

Again since horizontal distribution is integrable we haveA = 0 and putting it in (101) we obtain (111)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Uma Shankar Verma is thankful to University Grant Com-mission New Delhi India for financial support

References

[1] B OrsquoNeill ldquoThe fundamental equations of a submersionrdquo TheMichigan Mathematical Journal vol 13 pp 459ndash469 1966

[2] B OrsquoNeill Semi-Riemannian Geometry with Applications toRelativity vol 103 of Pure and Applied Mathematics AcademicPress New York NY USA 1983

[3] A Gray ldquoPseudo-Riemannian almost product manifolds andsubmersionsrdquo vol 16 pp 715ndash737 1967

[4] J P Bourguignon and H B Lawson A Mathematicians Visitto Kaluza-Klein Theory Rendiconti del Seminario Matematico1989

[5] S Ianus and M Visinescu ldquoSpace-time compactification andRiemannian submersionsrdquo in The Mathematical Heritage GRassias and C F Gauss Eds pp 358ndash371 World ScientificRiver Edge NJ USA 1991

[6] J P Bourguignon and H B Lawson Jr ldquoStability and isolationphenomena for Yang-Mills fieldsrdquo Communications in Mathe-matical Physics vol 79 no 2 pp 189ndash230 1981

[7] B Watson ldquoG 1198661015840

-Riemannian submersions and nonlineargauge field equations of general relativityrdquo in Global AnalysismdashAnalysis on Manifolds T Rassias and M Morse Eds vol 57 ofTeubner-Texte zur Mathematik pp 324ndash349 Teubner LeipzigGermany 1983

[8] C Altafini ldquoRedundant robotic chains on Riemannian submer-sionsrdquo IEEE Transactions on Robotics and Automation vol 20no 2 pp 335ndash340 2004

[9] M T Mustafa ldquoApplications of harmonic morphisms to grav-ityrdquo Journal of Mathematical Physics vol 41 no 10 pp 6918ndash6929 2000

[10] B Watson ldquoAlmost Hermitian submersionsrdquo Journal of Differ-ential Geometry vol 11 no 1 pp 147ndash165 1976

[11] D Chinea ldquoAlmost contact metric submersionsrdquo Rendiconti delCircolo Matematico di Palermo II vol 34 no 1 pp 319ndash3301984

[12] D Chinea ldquoTransference of structures on almost complexcontact metric submersionsrdquo Houston Journal of Mathematicsvol 14 no 1 pp 9ndash22 1988

[13] Y Gunduzalp and B Sahin ldquoPara-contact para-complexpseudo-Riemannian submersionsrdquo Bulletin of the MalaysianMathematical Sciences Society

[14] I Sato ldquoOn a structure similar to the almost contact structurerdquoTensor vol 30 no 3 pp 219ndash224 1976

[15] P K Rashevskij ldquoThe scalar field in a stratified spacerdquoTrudy Seminara po Vektornomu i Tenzornomu Analizu s ikhPrilozheniyami k Geometrii Mekhanike i Fizike vol 6 pp 225ndash248 1948

[16] P Libermann ldquoSur les structures presque paracomplexesrdquoComptes Rendus de lrsquoAcademie des Sciences I vol 234 pp 2517ndash2519 1952

[17] E M Patterson ldquoRiemann extensions which have Kahlermetricsrdquo Proceedings of the Royal Society of Edinburgh AMathematics vol 64 pp 113ndash126 1954

[18] S Sasaki ldquoOn differentiable manifolds with certain structureswhich are closely related to almost contact structure Irdquo TheTohoku Mathematical Journal vol 12 pp 459ndash476 1960

[19] S Zamkovoy ldquoCanonical connections on paracontact mani-foldsrdquo Annals of Global Analysis and Geometry vol 36 no 1pp 37ndash60 2009

[20] D E Blair Riemannian Geometry of Contact and SymplecticManifolds vol 23 of Progress in Mathematics BirkhauserBoston Mass USA 2002

[21] V Cruceanu P Fortuny and P M Gadea ldquoA survey onparacomplex geometryrdquoThe Rocky Mountain Journal of Mathe-matics vol 26 no 1 pp 83ndash115 1996

[22] P M Gadea and J M Masque ldquoClassification of almost para-Hermitian manifoldsrdquo Rendiconti di Matematica e delle sueApplicazioni VII vol 7 no 11 pp 377ndash396 1991

[23] M Falcitelli S Ianus and A M Pastore Riemannian Submer-sions and Related Topics World Scientific River Edge NJ USA2004

[24] E G Rio and D N Kupeli Semi-Riemannian Maps andTheir Applications Kluwer Academic Publisher DordrechtTheNetherlands 1999

[25] B Sahin ldquoSemi-invariant submersions from almost HermitianmanifoldsrdquoCanadianMathematical Bulletin vol 54 no 3 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 12: Research Article Paracomplex Paracontact Pseudo-Riemannian …downloads.hindawi.com/archive/2014/616487.pdf · 2019-07-31 · Research Article Paracomplex Paracontact Pseudo-Riemannian

12 Geometry

Theorem 32 Let 119891 119872119898

rarr 119872119899 be a paracomplex

paracontact pseudo-Riemannian submersion from a para-Kahler manifold119872 onto an almost paracontact pseudometricmanifold 119872 and let the fibres of 119891 be pseudo-Riemanniansubmanifolds of119872 If 119880 119883 are non-lightlike unit vertical andnon-lightlike unit horizontal vector fields respectively then onehas

119867(119880) = 119867V(119880) minus 2

1003817100381710038171003817T1198801198801003817100381710038171003817

2

(110)

119867(119883) = 119867 (119883) ∘ 119891 (111)

Proof Since 119891 is the paracomplex paracontact pseudo-Riemannian submersion from a para-Kahler manifold 119872

onto an almost paracontact pseudometric manifold 119872 by(16) and equations of Lemma 19 we have

119892 (T119869119880(119869119880) T

119880119880) = 119892 (119869

2

(T119880119880) T

119880119880) =

1003817100381710038171003817T1198801198801003817100381710038171003817

2

119892 (T119880(119869119880) T

119880(119869119880)) = minus119892 (T

119880119880T119880119880) = minus

1003817100381710038171003817T1198801198801003817100381710038171003817

2

(112)

and by using the above results in (100) we have

119867(119880) = 119867V(119880) minus

1003817100381710038171003817T1198801198801003817100381710038171003817

2

minus1003817100381710038171003817T119880119880

1003817100381710038171003817

2

= 119867V(119880) minus 2

1003817100381710038171003817T1198801198801003817100381710038171003817

2

(113)

Again since horizontal distribution is integrable we haveA = 0 and putting it in (101) we obtain (111)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Uma Shankar Verma is thankful to University Grant Com-mission New Delhi India for financial support

References

[1] B OrsquoNeill ldquoThe fundamental equations of a submersionrdquo TheMichigan Mathematical Journal vol 13 pp 459ndash469 1966

[2] B OrsquoNeill Semi-Riemannian Geometry with Applications toRelativity vol 103 of Pure and Applied Mathematics AcademicPress New York NY USA 1983

[3] A Gray ldquoPseudo-Riemannian almost product manifolds andsubmersionsrdquo vol 16 pp 715ndash737 1967

[4] J P Bourguignon and H B Lawson A Mathematicians Visitto Kaluza-Klein Theory Rendiconti del Seminario Matematico1989

[5] S Ianus and M Visinescu ldquoSpace-time compactification andRiemannian submersionsrdquo in The Mathematical Heritage GRassias and C F Gauss Eds pp 358ndash371 World ScientificRiver Edge NJ USA 1991

[6] J P Bourguignon and H B Lawson Jr ldquoStability and isolationphenomena for Yang-Mills fieldsrdquo Communications in Mathe-matical Physics vol 79 no 2 pp 189ndash230 1981

[7] B Watson ldquoG 1198661015840

-Riemannian submersions and nonlineargauge field equations of general relativityrdquo in Global AnalysismdashAnalysis on Manifolds T Rassias and M Morse Eds vol 57 ofTeubner-Texte zur Mathematik pp 324ndash349 Teubner LeipzigGermany 1983

[8] C Altafini ldquoRedundant robotic chains on Riemannian submer-sionsrdquo IEEE Transactions on Robotics and Automation vol 20no 2 pp 335ndash340 2004

[9] M T Mustafa ldquoApplications of harmonic morphisms to grav-ityrdquo Journal of Mathematical Physics vol 41 no 10 pp 6918ndash6929 2000

[10] B Watson ldquoAlmost Hermitian submersionsrdquo Journal of Differ-ential Geometry vol 11 no 1 pp 147ndash165 1976

[11] D Chinea ldquoAlmost contact metric submersionsrdquo Rendiconti delCircolo Matematico di Palermo II vol 34 no 1 pp 319ndash3301984

[12] D Chinea ldquoTransference of structures on almost complexcontact metric submersionsrdquo Houston Journal of Mathematicsvol 14 no 1 pp 9ndash22 1988

[13] Y Gunduzalp and B Sahin ldquoPara-contact para-complexpseudo-Riemannian submersionsrdquo Bulletin of the MalaysianMathematical Sciences Society

[14] I Sato ldquoOn a structure similar to the almost contact structurerdquoTensor vol 30 no 3 pp 219ndash224 1976

[15] P K Rashevskij ldquoThe scalar field in a stratified spacerdquoTrudy Seminara po Vektornomu i Tenzornomu Analizu s ikhPrilozheniyami k Geometrii Mekhanike i Fizike vol 6 pp 225ndash248 1948

[16] P Libermann ldquoSur les structures presque paracomplexesrdquoComptes Rendus de lrsquoAcademie des Sciences I vol 234 pp 2517ndash2519 1952

[17] E M Patterson ldquoRiemann extensions which have Kahlermetricsrdquo Proceedings of the Royal Society of Edinburgh AMathematics vol 64 pp 113ndash126 1954

[18] S Sasaki ldquoOn differentiable manifolds with certain structureswhich are closely related to almost contact structure Irdquo TheTohoku Mathematical Journal vol 12 pp 459ndash476 1960

[19] S Zamkovoy ldquoCanonical connections on paracontact mani-foldsrdquo Annals of Global Analysis and Geometry vol 36 no 1pp 37ndash60 2009

[20] D E Blair Riemannian Geometry of Contact and SymplecticManifolds vol 23 of Progress in Mathematics BirkhauserBoston Mass USA 2002

[21] V Cruceanu P Fortuny and P M Gadea ldquoA survey onparacomplex geometryrdquoThe Rocky Mountain Journal of Mathe-matics vol 26 no 1 pp 83ndash115 1996

[22] P M Gadea and J M Masque ldquoClassification of almost para-Hermitian manifoldsrdquo Rendiconti di Matematica e delle sueApplicazioni VII vol 7 no 11 pp 377ndash396 1991

[23] M Falcitelli S Ianus and A M Pastore Riemannian Submer-sions and Related Topics World Scientific River Edge NJ USA2004

[24] E G Rio and D N Kupeli Semi-Riemannian Maps andTheir Applications Kluwer Academic Publisher DordrechtTheNetherlands 1999

[25] B Sahin ldquoSemi-invariant submersions from almost HermitianmanifoldsrdquoCanadianMathematical Bulletin vol 54 no 3 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 13: Research Article Paracomplex Paracontact Pseudo-Riemannian …downloads.hindawi.com/archive/2014/616487.pdf · 2019-07-31 · Research Article Paracomplex Paracontact Pseudo-Riemannian

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Lagrangian Distributions and Connections in …forger/pdffiles/LDCSG.pdfcertain pseudo-riemannian manifolds, namely those of zero signature. However, the inter However, the inter-

Lagrangian Distributions and Connections in …forger/pdffiles/LDCSG.pdfcertain pseudo-riemannian manifolds, namely those of zero signature. However, the inter However, the inter-

WEAKLY SYMMETRIC PSEUDO RIEMANNIAN NILMANIFOLDSjawolf/publications.pdf/paper_192.pdf · Weakly symmetric spaces, introduced by A. Selberg [16], play important roles in number theory,

WEAKLY SYMMETRIC PSEUDO RIEMANNIAN NILMANIFOLDSjawolf/publications.pdf/paper_192.pdf · Weakly symmetric spaces, introduced by A. Selberg [16], play important roles in number theory,

Fundamental Groups of Flat Pseudo-Riemannian …References O. Baues,Prehomogeneous A ne Representations and Flat Pseudo-Riemannian Manifolds, in ‘Handbook of Pseudo-Riemannian Geometry’,

Fundamental Groups of Flat Pseudo-Riemannian …References O. Baues,Prehomogeneous A ne Representations and Flat Pseudo-Riemannian Manifolds, in ‘Handbook of Pseudo-Riemannian Geometry’,

Hilbert Space and pseudo-Riemannian Space: The Common Base of Quantum Information

Hilbert Space and pseudo-Riemannian Space: The Common Base of Quantum Information

Differential geometry Lecture 12: Pseudo-Riemannian manifolds

Differential geometry Lecture 12: Pseudo-Riemannian manifolds

Riemannian curvature of the NC 3-spherehep.itp.tuwien.ac.at/~miw/bzell2015/Arnlind_2015.pdf · Introduction Pseudo-Riemannian calculus NC 3-sphere Summary Introduction We are interested

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Introduction to Riemannian and Sub-Riemannian …Introduction to Riemannian and Sub-Riemannian geometry FromHamiltonianviewpoint andrei agrachev davide barilari ugo boscain This version:

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PSEUDO HARMONIC MORPHISMS ON RIEMANNIAN POLYHEDRAusers.ictp.it/~pub_off/preprints-sources/2004/IC2004102P.pdf · The outline of the paper is as follows. In Section 2 we recall some

PSEUDO HARMONIC MORPHISMS ON RIEMANNIAN POLYHEDRAusers.ictp.it/~pub_off/preprints-sources/2004/IC2004102P.pdf · The outline of the paper is as follows. In Section 2 we recall some

Recent Development s in Pseudo-Riemannian Geometrycharette/recherche/Primer.pdf · 0 is the usual constant curv ature Riemannian metric on S 3. The conformal transformations preserv

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I. GENERAL RELATIVITY { A SUMMARY A. Pseudo …jo/quantum/qm/gw2/Notes.pdf · I. GENERAL RELATIVITY { A SUMMARY A. Pseudo-Riemannian manifolds Spacetime is a manifold that is continuous

I. GENERAL RELATIVITY { A SUMMARY A. Pseudo …jo/quantum/qm/gw2/Notes.pdf · I. GENERAL RELATIVITY { A SUMMARY A. Pseudo-Riemannian manifolds Spacetime is a manifold that is continuous

Acoustic Torsion - Fields, Vacuum, And the Mirror Universe - Four-dimensional Pseudo-Riemannian Space

Acoustic Torsion - Fields, Vacuum, And the Mirror Universe - Four-dimensional Pseudo-Riemannian Space

DEPARTMENT OF MATHEMATICS, STATISTICS, AND PHYSICS · 2018-07-22 · and calculus of variations), Differential Geometry and Mathematical Physics (pseudo-Riemannian manifolds, geometric

DEPARTMENT OF MATHEMATICS, STATISTICS, AND PHYSICS · 2018-07-22 · and calculus of variations), Differential Geometry and Mathematical Physics (pseudo-Riemannian manifolds, geometric

Ergodic Theory and Dynamical Systemsperso.ens-lyon.fr/zeghib/Stationnary.pdfLorentzian manifolds, i.e. manifolds endowed with metric tensors of index 1, play a special role in pseudo-Riemannian

Ergodic Theory and Dynamical Systemsperso.ens-lyon.fr/zeghib/Stationnary.pdfLorentzian manifolds, i.e. manifolds endowed with metric tensors of index 1, play a special role in pseudo-Riemannian

LTH 741 - arXiv · LTH 741 hep-th/0703037 Special Geometry, Black Holes and Euclidean Supersymmetry ... ∗Contribution to the Handbook on Pseudo-Riemannian Geometry and Supersymmetry,

LTH 741 - arXiv · LTH 741 hep-th/0703037 Special Geometry, Black Holes and Euclidean Supersymmetry ... ∗Contribution to the Handbook on Pseudo-Riemannian Geometry and Supersymmetry,

Scienti c Report for 2008 - esi.ac.at · vanni Gallavotti, Wolfgang L. Reiter and Jakob Yngvason, 350 p.), and Recent Developments in Pseudo-Riemannian Geometry (edited by Dmitry

Scienti c Report for 2008 - esi.ac.at · vanni Gallavotti, Wolfgang L. Reiter and Jakob Yngvason, 350 p.), and Recent Developments in Pseudo-Riemannian Geometry (edited by Dmitry

Isometric immersions of space forms into Riemannian and ...lymber/files/textos/RMS_56_3_R01.pdfIsometric immersions of space forms into Riemannian and pseudo-Riemannian spaces of constant

Isometric immersions of space forms into Riemannian and ...lymber/files/textos/RMS_56_3_R01.pdfIsometric immersions of space forms into Riemannian and pseudo-Riemannian spaces of constant

Recent developments in pseudo-Riemannian holonomy theory · Recent developments in pseudo-Riemannian holonomy theory Thomas Leistner Humboldt University Berlin Minisymposium Differential

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Homomorphisms of pseudo-Riemannian calculi and …liu_se_01.pdf · Workshop on Quantum Geometry, Field Theory and Gravity Corfu, 2019. Introduction Pseudo-Riemannian calculi Examples

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Riemannian Geometry - univie.ac.atcap/files/Riemannian-1.pdf · CHAPTER 1 Fundamentals of Riemannian geometry After recalling some background, we de ne Riemannian metrics and Riemannian

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On Riemannian geometry of orthonormal frame bundles · c on OM defined depending on a nonzero constant c, which are Riemannian for c>0 and pseudo-Riemannian for c

On Riemannian geometry of orthonormal frame bundles · c on OM defined depending on a nonzero constant c, which are Riemannian for c>0 and pseudo-Riemannian for c