Research Article On Some Transverse Geometrical Structures...
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Research Article On Some Transverse Geometrical Structures of Lifted Foliation to Its Conormal Bundle Cristian Ida and Alexandru Oans Department of Mathematics and Computer Science, Transilvania University of Bras ¸ov, Street Iuliu Maniu 50, 500091 Bras ¸ov, Romania Correspondence should be addressed to Cristian Ida; [email protected] Received 15 October 2014; Revised 9 February 2015; Accepted 20 February 2015 Academic Editor: Peter G. L. Leach Copyright © 2015 C. Ida and A. Oan˘ a. 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 consider the liſt of a foliation to its conormal bundle and some transverse geometrical structures associated with this foliation are studied. We introduce a good vertical connection on the conormal bundle and, moreover, if the conormal bundle is endowed with a transversal Cartan metric, we obtain that the liſted foliation to its conormal bundle is a Riemannian one. Also, some transversally framed (3, )-structures of corank 2 on the normal bundle of liſted foliation to its conormal bundle are introduced and an almost (para)contact structure on a transverse Liouville distribution is obtained. 1. Introduction and Preliminaries e study of the liſt of transversal Finsler foliations to their normal bundle using the technique of good vertical connection was initiated by Miernowski and Mozgawa [1] where it is proved that the liſted foliation is a Riemannian one. Also, using different methods, some connections between foliations and Lagrangians (or Hamiltonians) in order to recover Riemannian foliations are investigated in the recent papers [2–5]. Our aim in this paper is to extend the study from [1] for the case of liſted foliation to its conormal bundle. In this sense we introduce a good vertical connection on the conormal bundle and we give an application of it in order to obtain that the liſted foliation is a Riemannian one in the case when the conormal bundle is endowed with a transversal Cartan metric. Moreover, in this case, some transversally framed (3, )-structures and an almost (para)contact structure associated with liſted foliation are investigated. e methods used here are similarly and closely related to those used in [1, 6] for the case of transversal Finsler foliations. Let us consider an (+)-dimensional manifold which will be assumed to be connected and orientable. Definition 1. A codimension foliation F on is defined by a foliated cocycle { , , , } such that (i) { }, ∈, is an open covering of ; (ii) for every ∈, : → are submersions, where is an -dimensional manifold, called transversal manifold; (iii) the maps , : ( ∩ )→ ( ∩ ) satisfy = , ∘ (1) for every (, ) ∈ × such that ∩ ̸ =0. Every fibre of is called a plaque of the foliation. Condition (1) says that on the intersection ∩ the plaques defined, respectively, by and coincide. e manifold is decomposed into a family of disjoint immersed connected submanifolds of dimension ; each of these submanifolds is called a leaf of F. By F we denote the tangent bundle to F and Γ(F) is the space of its global sections, that is, vector fields tangent to F, and by F = /F we denote the normal bundle of F. In this paper, a system of local coordinates adapted to the foliation F means coordinates ( 1 ,..., , 1 ,..., ) on an open subset on which the foliation is trivial and defined by the equations = 0, = 1, . . . , . We notice that the total spaces of the conormal bundle ∗ F of F carry a natural foliation F of codimension 2 such Hindawi Publishing Corporation e Scientific World Journal Volume 2015, Article ID 218912, 6 pages http://dx.doi.org/10.1155/2015/218912
Transcript of Research Article On Some Transverse Geometrical Structures...
Research ArticleOn Some Transverse Geometrical Structures of LiftedFoliation to Its Conormal Bundle
Cristian Ida and Alexandru Oans
Department of Mathematics and Computer Science Transilvania University of Brasov Street IuliuManiu 50 500091 Brasov Romania
Correspondence should be addressed to Cristian Ida cristianidaunitbvro
Received 15 October 2014 Revised 9 February 2015 Accepted 20 February 2015
Academic Editor Peter G L Leach
Copyright copy 2015 C Ida and A Oana This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
We consider the lift of a foliation to its conormal bundle and some transverse geometrical structures associatedwith this foliation arestudied We introduce a good vertical connection on the conormal bundle and moreover if the conormal bundle is endowed witha transversal Cartan metric we obtain that the lifted foliation to its conormal bundle is a Riemannian one Also some transversallyframed 119891(3 120576)-structures of corank 2 on the normal bundle of lifted foliation to its conormal bundle are introduced and an almost(para)contact structure on a transverse Liouville distribution is obtained
1 Introduction and Preliminaries
The study of the lift of transversal Finsler foliations totheir normal bundle using the technique of good verticalconnection was initiated by Miernowski and Mozgawa [1]where it is proved that the lifted foliation is a Riemannian oneAlso using different methods some connections betweenfoliations and Lagrangians (or Hamiltonians) in order torecover Riemannian foliations are investigated in the recentpapers [2ndash5] Our aim in this paper is to extend the studyfrom [1] for the case of lifted foliation to its conormal bundleIn this sense we introduce a good vertical connection onthe conormal bundle and we give an application of it inorder to obtain that the lifted foliation is a Riemannianone in the case when the conormal bundle is endowedwith a transversal Cartan metric Moreover in this casesome transversally framed 119891(3 120576)-structures and an almost(para)contact structure associated with lifted foliation areinvestigated
The methods used here are similarly and closely relatedto those used in [1 6] for the case of transversal Finslerfoliations
Let us consider119872 an (119899+119898)-dimensionalmanifoldwhichwill be assumed to be connected and orientable
Definition 1 A codimension 119899 foliationF on119872 is defined bya foliated cocycle 119880
119894 120593119894 119891119894119895 such that
(i) 119880119894 119894 isin 119868 is an open covering of119872
(ii) for every 119894 isin 119868 120593119894 119880119894rarr 119873 are submersions where
119873 is an 119899-dimensional manifold called transversalmanifold
(iii) the maps 119891119894119895
120593119894(119880119894cap 119880119895) rarr 120593
119895(119880119894cap 119880119895) satisfy
120593119895= 119891119894119895∘ 120593119894 (1)
for every (119894 119895) isin 119868 times 119868 such that 119880119894cap 119880119895
= 0
Every fibre of 120593119894is called a plaque of the foliation
Condition (1) says that on the intersection119880119894cap119880119895the plaques
defined respectively by 120593119894and 120593
119895coincide The manifold119872
is decomposed into a family of disjoint immersed connectedsubmanifolds of dimension 119898 each of these submanifolds iscalled a leaf ofF
By 119879F we denote the tangent bundle to F and Γ(F) isthe space of its global sections that is vector fields tangent toF and by 119876F = 119879119872119879F we denote the normal bundle ofF
In this paper a system of local coordinates adapted to thefoliationFmeans coordinates (1199091 119909119899 1199101 119910119898) on anopen subset119880 on which the foliation is trivial and defined bythe equations 119889119909119886 = 0 119886 = 1 119899
We notice that the total spaces of the conormal bundle119876lowastF ofF carry a natural foliation F of codimension 2119899 such
Hindawi Publishing Corporatione Scientific World JournalVolume 2015 Article ID 218912 6 pageshttpdxdoiorg1011552015218912
2 The Scientific World Journal
that the leaves of F are covering spaces of the leaves ofF andit is called the natural lift ofF to its conormal bundle 119876lowastF
If we denote by 119889119909119886 119886 = 1 119899 the correspondinglocal coframe on119876lowastF thenwe can induce a chart (119909119886 119901
119886 119910119906)
on 119876lowastF where 119901 = 119901119886119889119909119886 isin Γ(119876lowastF) and the system of
equations 119909119886 = const 119901119886= const defines the foliation F
Let 119876F = 119879(119876lowastF)119879F be the normal bundle ofthe foliated manifold (119876lowastF F) The vectors 120597120597119909119886 120597120597119901
119886
119886 = 1 119899 form a natural frame of 119876F at the point(119909119886 119901
119886 119910119906) isin 119876lowastF The canonical projection 120587 119876lowastF rarr
119872 given by 120587(119909119886 119901119886 119910119906) = (119909119886 119910119906) induces another projec-
tion 120587lowast 119879(119876lowastF) rarr 119879119872 which maps the vectors tangent
to F in the vectors tangent toFThus 120587lowastinduces amapping
lowast
119876F rarr 119876F and is denoted by 119881(119876lowastF) = kerlowast
which is a vertical bundle spanned by the vectors 120597120597119901119886
119886 = 1 119899
Lemma 2 Let 119900 119872 rarr 119876lowastF be the zero section of theconormal bundle119876lowastFThen the set 119900(119872) is saturated on119876lowastFwith foliation F
2 Good Vertical Connection on (119876lowastFF)
The purpose of this section is to define a linear connectionnabla X(119881(119876lowastF)) rarr X(Tlowast(119876lowastF) otimes 119881(119876lowastF)) related toconsidered foliated structure where 119876lowastF = 119876lowastF minus 119900(119872)Sincewe have the foliatedmanifold (119876lowastF F) we are lookingfor a Bott connection such that for any vector field119883 tangentto F and any transversal vector field 119884 we have
nabla119883119884 = 119901
119876F ([119883 ]) (2)
where 119901119876F 119879(119876lowastF) rarr 119876F is the canonical projection
and 119901119876F() = 119884
Let us consider now the F-transversal Hamilton-Liou-ville vector field defined by 119862lowast 119876lowastF rarr 119881(119876lowastF)119862lowast(119909119886 119901
119886 119910119906) = 119901
119886(120597120597119901119886) It can be checked that this defini-
tion is well posed From the definition of the Bott connectionthe following lemma holds
Lemma 3 Letnabla X(119881(119876lowastF)) rarr X(119879lowast(119876lowastF)otimes119881(119876lowastF))be a Bott connection Then nabla
119883119862lowast = 0 for every vector field
tangent to F
Now consider the local frame 120597120597119909119886 120597120597119901119886 120597120597119910119906 of
119879(119876lowastF) and recall that the vectors 120597120597119901119886 form the basis of
119881(119876lowastF) With these settings we put
nabla120597120597119909119886
120597
120597119901119887
= Γ119887119886119888
120597
120597119901119888
nabla120597120597119901119886
120597
120597119901119887
= Γ119886119887119888
120597
120597119901119888
nabla120597120597119910119906
120597
120597119901119887
= Γ119887119906119888
120597
120597119901119888
(3)
From the above formulas it follows that
Γ119887119906119888
= 0 nabla120597120597119901119886
119862lowast = (120575119886119888+ 119901119887Γ119886119887119888)
120597
120597119901119888
(4)
The Bott connection nabla allows us to define a mapping
119871 X (119876F) 997888rarr X (119881 (119876lowastF)) 119871 (119883) = nabla119862lowast (5)
where 119901119876F(119883) = 119883 If we denote by Λ the restriction of the
linear mapping 119871 to the bundle 119881(119876lowastF) then we can statethe following
Definition 4 The Bott connection nabla is said to be a goodvertical connection if Λ 119881(119876lowastF) rarr 119881(119876lowastF) is a bundleisomorphism
Observe thatnabla is a good vertical connection if and only ifthematrix 120575119886
119888+119901119887Γ119886119887119888
is nondegenerated If we put119867(119876lowastF) =
ker119871 then we can split the bundle 119876F into direct sum
119876F = 119867 (119876lowastF) oplus 119881 (119876lowastF) (6)
The coefficients of the mapping 119871 in the basis 120597120597119909119886 120597120597119901119886
of 119876F are
119871(120597
120597119909119886) = 119901
119887Γ119887119886119888
120597
120597119901119888
119871 (120597
120597119901119886
) = (120575119886119888+ 119901119887Γ119886119887119888)
120597
120597119901119888
= 119871119886119888
120597
120597119901119888
(7)
It is easy to check that the vectors 120575120575119909119886 = 120597120597119909119886 +
119873119886119887(120597120597119901119887) where119873
119886119887= minus(119871minus1)119888
119887119901119889Γ119889119886119888 form a basis of ker119871
In the sequel we will use the basis 120575120575119909119886 120597120597119901119886 called
adapted as well as its dual 119889119909119886 120575119901119886= 119889119901119886minus 119873119886119887119889119909119887 Using
this coframe we can define the local connection forms by
nabla120597
120597119901119887
= 120596119887119886otimes
120597
120597119901119886
(8)
where
120596119887119886= Γ119887119888119886119889119909119888 + Γ119887119888
119886119889119901119888= (Γ119887119888119886+ Γ119887119889119886119873119889119888) 119889119909119888 + Γ119887119888
119886120575119901119888
= 119867119887119888119886119889119909119888 + Γ119887119888
119886120575119901119888
(9)
Notice that 119867119887119888119886119901119887
= 119873119888119886 The formula 120579(120597120597119901
119886) = 120575120575119909119886
defines a linear mapping 120579 119881(119876lowastF) rarr 119867(119876lowastF)This mapping allows us to extend the connection nabla to thehorizontal bundle119867(119876lowastF) by
nabla119883119884 = 120579 (nabla
119883120579minus1 (119884)) (10)
where 119884 isin Γ(119867(119876lowastF)) 119883 isin Γ(119879(119876lowastF)) In this way weconstruct a linear connection in 119876F
nabla119883119884 = nabla
119883 (V (119884)) + nabla119883 (119884 minus V (119884)) (11)
where 119884 isin Γ(119876F) 119883 isin Γ(119879(119876lowastF)) and V 119876F rarr119881(119876lowastF) is the vertical projection from decomposition (6)In particular we have
nabla120575
120575119909119886= 120596119887119886otimes
120575
120575119909119887 (12)
where 120596119887119886is given in (9)
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If 120593 isin Γ(119876lowastF otimes 119876F) is a 1-form with values in 119876Flocally given by
120593 = 120593119886 otimes120575
120575119909119886+ 120593119887otimes
120597
120597119901119887
(13)
then following [1 7] we can define an exterior differential119863120593 by putting
119863120593 = (119889120593119886 minus 120593119886 and 120596119888119886) otimes
120575
120575119909119886+ (119889120593
119887minus 120593119887and 120596119887119888) otimes
120597
120597119901119888
(14)
A straightforward calculus shows that the above formula iswell defined
The bundle 119876lowastF otimes 119876F admits a natural section 120578 givenby
120578 = 119889119909119886 otimes120597
120597119909119886+ 119889119901119887otimes
120597
120597119901119887
= 119889119909119886 otimes120575
120575119909119886+ 120575119901119887otimes
120597
120597119901119887
(15)
It is clear that the form 120578 is well defined
Definition 5 The form 120577 = 119863120578 is called the torsion form ofthe connection nabla
Locally the form 120577 can be expressed as follows
119863120578 = (minus119889119909119886 and 120596119888119886) otimes
120575
120575119909119888+ (119889 (120575119901
119887) minus 120575119901
119888and 120596119888119887) otimes
120597
120597119901119887
= 120577119888 otimes120575
120575119909119888+ 120577119887otimes
120597
120597119901119887
(16)
where
120577119888 =1
2(119867119888119890119886minus 119867119888119886119890) 119889119909119886 and 119889119909119890 minus Γ119888119890
119886119889119909119886 and 120575119901
119890
120577119887= minus119889119873
119888119887and 119889119909119888 minus 119867119886
119890119887120575119901119886and 119889119909119890 minus Γ119886119890
119887120575119901119886and 120575119901119890
(17)
3 Transversal Cartan Metrics on 119876lowastF andRiemannian Foliations
As in the case of transversal Finsler metrics on the normalbundle of a foliation [1 3] a transversal Cartan metric on119876lowastF is a basic function (with respect to the lifted foliationF) 119870 119876lowastF rarr [0infin) which has the following properties
(i) 119870 is 119862infin on 119876lowastF(ii) 119870(119909 120582119901) = 120582119870(119909 119901) for all 120582 gt 0(iii) the 119899 times 119899 matrix (119892119886119887) where 119892119886119887 =
(12)(12059721198702120597119901119886120597119901119886) is positive definite at all
points of 119876lowastFAlso 119870(119909 119901) gt 0 whenever 119901 = 0 As usual [8] the proper-ties of119870 imply that
119901119886 = 119892119886119887119901119887 119901
119886= 119892119886119887119901119887 1198702 = 119892119886119887119901
119886119901119887= 119901119886119901119886
119862119886119887119888119901119888= 119862119886119888119887119901
119888= 119862119888119886119887119901
119888= 0
(18)
where (119892119886119887) is the inverse matrix of (119892119887119886) and we have put
119901119886 = (12)(1205971198702120597119901119886) 119862119886119887119888 = minus(14)(12059731198702120597119901
119886120597119901119887120597119901119888)
Also 119892119886119887 determines a metric structure on 119881(119876lowastF) bysetting
119866V(119883 119884) = 119892119886119887 (119909 119901)119883
119886(119909 119910 119901) 119884
119887(119909 119910 119901) (19)
for every 119883 = 119883119886(119909 119910 119901)(120597120597119901
119886) and 119884 = 119884
119887(119909 119910 119901)(120597
120597119901119887) isin Γ(119881(119876lowastF))Similar reasons as for transversal Finsler foliations (see
Theorem 31 from [1]) lead to the following result
Theorem 6 Let 119870 119876lowastF rarr [0infin) be a transversal Cartanmetric and let 119866V be the Riemannian metric on 119881(119876lowastF)induced by 119870 as in (19) Then there exists exactly one Bottvertical connection nabla X(119881(119876lowastF)) rarr X(119879lowast(119876lowastF) otimes119881(119876lowastF)) such that
(i) nabla is a good vertical connection
(ii) if119883119884 isin Γ(119881(119876lowastF)) and 119885 isin Γ(119879(119876lowastF)) then
119885119866V(119883 119884) = 119866V (nabla
119885119883119884) + 119866V (119883 nabla
119885119884) (20)
(iii) 120577(119883 119884) = 0 for every119883119884 isin Γ(119881(119876lowastF))
(iv) 120577(119883 119884) isin Γ(119881(119876lowastF)) for every119883119884 isin Γ(119867(119876lowastF))
Also the isomorphism 120579 does not depend on the coordi-nates along the leaves of F so the Riemannianmetric in119876F
defined by119866 = 119866ℎ+119866V where119866ℎ(119883 119884) = 119866V(120579minus1(119883) 120579minus1(119884))for every 119883119884 isin Γ(119867(119876lowastF)) and 119866(119883 119884) = 0 for every119883 isin Γ(119867(119876lowastF)) and 119884 isin Γ(119881(119876lowastF)) is a transversalRiemannian metric for the lifted foliation F to the conormalbundle 119876lowastF ofF Hence we can consider the following
Theorem 7 If the conormal bundle of foliationF is endowedwith a transversal Cartan metric then the lifted foliation F tothe conormal bundle 119876lowastF is Riemannian
4 Transversally Framed119891(3120576)-Structures on (119876lowastFF)
The study of structures on manifolds defined by a tensorfield satisfying 1198913 plusmn 119891 = 0 has the origin in a paper byYano [9] Later on these structures have been genericallycalled 119891-structures On the tangent manifold of a Finslerspace the notion of framed119891(3 1)-structure was defined andstudied by Anastasiei in [10] and on the cotangent bundle ofa Cartan space the study is continued in [11 12] Taking intoaccount that the conormal bundle 119876lowastF has a local modelof a cotangent manifold in this section we extend the studyconcerning 119891-structures in our context
Let 120576 = plusmn1 A framed 119891(3 120576)-structure of corank 119904 on a(2119899 + 119904)-dimensional manifold 119873 is a natural generalizationof an almost contact structure on 119873 (for 120576 = 1) and of analmost paracontact structure on119873 (for 120576 = minus1) respectivelyand it is a triplet (119891 (120585
119894) (120596119894)) 119894 = 1 119904 where 119891 is a tensor
4 The Scientific World Journal
field of type (1 1) (120585119894) are vector fields and (120596119894) are 1-forms
on119873 such that
120596119894 (120585119895) = 120575119894119895 119891 (120585
119894) = 0 120596119894 ∘ 119891 = 0
1198912 = minus120576(119868 minussum119894
120596119894 otimes 120585119894)
(21)
where 119868 denotes the Kronecker tensor field on 119873 The nameof 119891(3 120576)-structure was suggested by the identity1198913 +120576119891 = 0For an account of such kind of structures we refer to [13]
Let us consider now that 119876lowastF is endowed with atransversal Cartan metric 119870 The linear operator 120601 given inthe local adapted basis of 119876F by
120601(120575
120575119909119886) = minus120576119892
119886119887
120597
120597119901119887
120601 (120597
120597119901119886
) = 119892119886119887120575
120575119909119887(22)
defines an almost complex structure on 119876F for 120576 = 1 and analmost paracomplex structure on119876F for 120576 = minus1 respectivelyWe also have
119866 (120601 (119883) 120601 (119884)) = 119866 (119883 119884) forall119883 119884 isin Γ (119876F) (23)
Let us put 1205851= (119901119886119870)(120575120575119909119886) and 120585
2= (119901119886119870)(120597120597119901
119886) =
(1119870)119862lowast Thus we have two global transverse vector fieldson (119876lowastF F) which are linearly independent The first istransversally horizontal and the second one is transversallyvertical
From the definition of 120601 it follows
120601 (1205851) = minus120576120585
2 120601 (120585
2) = 1205851 (24)
Now if we consider the dual transverse 1-forms of 1205851and
1205852 respectively locally given by 1205961 = (119901
119886119870)119889119909119886 and 1205962 =
(119901119886119870)120575119901119886 then we easily check that
1205961 ∘ 120601 = 1205962 1205962 ∘ 120601 = minus1205761205961
1205961 (119883) = 119866 (119883 1205851) 1205962 (119883) = 119866 (119883 120585
2) forall119883 isin Γ (119876F)
(25)
Next using120601 120585119894 and120596119894 119894 isin 1 2 we construct the transverse
tensor field 119891 of type (1 1) on (119876lowastF F) by putting
119891 (119883) = 120601 (119883) minus 1205962 (119883) 1205851 + 1205761205961 (119883) 1205852 119883 isin Γ (119876F)
(26)
Using a similar argument as in [6 10ndash12] by direct calculuswe obtain the following
Theorem 8 The triple (119891 (120585119894) (120596119894)) 119894 isin 1 2 provides
some transversally framed 119891(3 120576)-structures of corank 2 on(119876lowastF F) that is the following hold
(i) 120596119894(120585119895) = 120575119894119895 119891(120585119894) = 0 120596119894 ∘ 119891 = 0
(ii) 1198912 = minus120576(119868 minus 1205961 otimes 1205851minus 1205962 otimes 120585
2)
(iii) 119891 is of rank 2119899 minus 2 and 1198913 + 120576119891 = 0
Theorem 9 The transversal Riemannian metric 119866 on(119876lowastF F) satisfies
119866 (119891 (119883) 119891 (119884)) = 119866 (119883 119884) minus 1205961 (119883) 1205961(119884)
minus 1205962 (119883) 1205962(119884) 119883 119884 isin Γ (119876F)
(27)
For 120576 = 1 we put
Φ (119883 119884) = 119866 (119891 (119883) 119884) 119883 119884 isin Γ (119876F) (28)
By usingTheorems 8 and 9 we obtain
Φ (119883 119884) = minusΦ (119884119883) 119883 119884 isin Γ (119876F) (29)
ThusΦ is a transverse 2-form on (119876lowastF F) It is degeneratewith null space span120585
1 1205852
Also using the calculus in local coordinates we easilyobtain
Φ(120575
120575119909119886
120575
120575119909119887) = 0 Φ(
120575
120575119909119886
120597
120597119901119887
) = minus120575119887119886+119901119886119901119887
1198702
Φ (120597
120597119901119886
120597
120597119901119887
) = 0
(30)
On the other hand we have
1198891205961 (120575
120575119909119886
120575
120575119909119887) =
120575
120575119909119886(119901119887
119870) minus
120575
120575119909119887(119901119886
119870)
1198891205961 (120575
120575119909119886
120597
120597119901119887
) =1
119870(minus120575119887119886+119901119886119901119887
1198702)
1198891205961 (120597
120597119901119886
120597
120597119901119887
) = 0
(31)
where in the second relation we have used 119901119886 = 119870(120597119870120597119901119886)
which follows from the homogeneity conditions (18) of 119870Comparing now Φ with 1198891205961 we obtain
1
119870Φ = 1198891205961 + Ψ (32)
where Ψ = ((120575120575119909119887)(119901119886119870) minus (120575120575119909119886)(119901
119887119870))119889119909119886 and 119889119909119887
Thus (1119870)Φ is transversally closed if and only if Ψ istransversally closed Concluding (1119870)Φ is in general analmost transversally presymplectic structure on (119876lowastF F)
Similarly for 120576 = minus1 we can put
119867(119883 119884) = 119866 (119891 (119883) 119884) 119883 119884 isin Γ (119876F) (33)
We have the following
Theorem 10 The mapping119867 is a symmetric bilinear form on(119876lowastF F) and the annihilator of119867 is ker119891
The Scientific World Journal 5
Proof The symmetry and bilinearity are obvious Also thenull space of119867 is
119883 isin Γ (119876F) | 119867 (119883 119884) = 0 forall119884 isin Γ (119876F)
= 119883 isin Γ (119876F) | 119866 (119891 (119883) 119884) = 0 = ker119891(34)
which end the proof
Locally we obtain
119867 = (119892119886119887
minus119901119886119901119887
1198702)119889119909119886 otimes 119889119909119887 minus (119892119886119887 minus
119901119886119901119887
1198702)120575119901119886otimes 120575119901119887
(35)
with det(119892119886119887
minus (1199011198861199011198871198702)) = 0 since (119892
119886119887minus (1199011198861199011198871198702))119901119887 =
119901119886minus 119901119886= 0 and similarly det(119892119886119887 minus (1199011198861199011198871198702)) = 0 since
(119892119886119887 minus (1199011198861199011198871198702))119901119887= 119901119886 minus 119901119886 = 0
Remark 11 The map 119867 is a transversally singular pseudo-Riemannian metric on (119876lowastF F)
5 An Almost (Para)Contact Structure onTransverse Liouville Distribution of(119876lowastFF)
Denote by 1205852 the line vector bundle over119876lowastF spanned by 120585
2
and we define the transverse vertical Liouville distribution asthe complementary orthogonal distribution 119878(119876lowastF) to 120585
2
in119881(119876lowastF)with respect to119866V namely119881(119876lowastF) = 119878(119876lowastF)oplus
1205852 Hence 119878(119876lowastF) is defined by 120572 = 1205962|
119881(119876lowastF) that is
Γ (119878 (119876lowastF)) = 119883 isin Γ (119881 (119876lowastF)) 120572 (119883) = 0 (36)
Thus any transverse vertical vector field119883 isin Γ(119881(119876lowastF)) canbe expressed as
119883 = 119875119883 + 120572 (119883) 1205852 (37)
where 119875 is the projection morphism of 119881(119876lowastF) on 119878(119876lowastF)By direct calculations one gets the following
Proposition 12 For any transverse vertical vector fields119883119884 isinΓ(119881(119876lowastF)) one has
119866V(119883 119875119884) = 119866V
(119875119883 119875119884) = 119866V(119883 119884) minus 120572 (119883) 120572 (119884) (38)
Theorem 13 The transverse vertical Liouville distribution119878(119876lowastF) is integrable
Proof The proof follows using an argument similar to Theo-rem 31 [14] (see alsoTheorem 21 [6] orTheorem 4 [15])
In the following we will consider the transverse Liouvilledistribution of (119876lowastF F) as the complementary orthogonaldistribution 119878(119876lowastF) to 120585
2 in 119876F with respect to 119866 that is
119878(119876lowastF) = 119867(119876lowastF) oplus 119878(119876lowastF)Let us restrict to 119878(119876lowastF) all the geometrical structures
introduced in Section 4 for all 119876F We indicate this byoverlines Hence we have
(i) 1205851= 1205851since 120585
1lies in 119878(119876lowastF)
(ii) 1205962 = 0 since 1205962(119883) = 119866(119883 1205852) = 0 for every
transverse vector field119883 isin 119878(119876lowastF)(iii) 119866 = 119866|
119878(119876lowastF)
(iv) 119891(119883) = 120601(119883) + 1205761205961(119883) otimes 1205852is an endomorphism of
119878(119876lowastF) since
119866(119891 (119883) 1205852) = 119866 (120601 (119883) 1205852) + 1205761205961 (119883)119866 (1205852 1205852)
= 1205962 (120601 (119883)) + 1205761205961 (119883) = 0(39)
We denote now 120585 = 1205851and 120578 = 1205961
By Theorem 8 we obtain the following
Theorem 14 The triple (119891 120585 120578) provides an almost(para)contact structure on 119878(119876lowastF) that is
(i) 1198913
+ 120576119891 = 0 119903119886119899119896119891 = 2119899 minus 2 = (2119899 minus 1) minus 1
(ii) 120578(120585) = 1 119891(120585) = 0 120578 ∘ 119891 = 0
(iii) 1198912
(119883) = minus120576(119883 minus 120578(119883)120585) for119883 isin 119878(119876lowastF)
Also by Theorem 9 we obtain the following
Theorem 15 The transversal Riemannian metric 119866 verifies
119866(119891 (119883) 119891 (119884)) = 119866 (119883 119884) minus 120578 (119883) 120578 (119884) (40)
for every transverse vector fields119883119884 isin 119878(119876lowastF)
Concluding the ensemble (119891 120585 120578 119866) is an almost(para)contact Riemannian structure on 119878(119876lowastF)
Conflict of Interests
The authors declare that they have no conflict of interests
Authorsrsquo Contribution
Both authors contributed equally to the paper All the authorsread and approved the final paper
Acknowledgment
The first author is supported by the Grant ldquoFellowship ofTransilvania University of Brasov 2014rdquo
References
[1] A Miernowski andWMozgawa ldquoLift of the Finsler foliation toits normal bundlerdquo Differential Geometry and Its Applicationsvol 24 no 2 pp 209ndash214 2006
[2] P Popescu and C Ida ldquoFoliations compatible with hamilto-niansrdquo in Proceedings of the BSG International Conference onDifferential Geometry and Dynamical Systems (DGDS rsquo13) pp147ndash155 Bucharest Romania October 2013
6 The Scientific World Journal
[3] P Popescu and M Popescu ldquoLagrangians adapted to submer-sions and foliationsrdquoDifferential Geometry and Its Applicationsvol 27 no 2 pp 171ndash178 2009
[4] P Popescu and M Popescu ldquoFoliated vector bundles andRiemannian foliationsrdquo Comptes Rendus Mathematique vol349 no 7-8 pp 445ndash449 2011
[5] P Popescu and M Popescu ldquoLagrangians and Hamiltonianstructures and foliationsrdquo Physics AUC vol 21 no 1 pp 203ndash206 2011
[6] C Ida ldquoSome framed 119891-structures on transversally Finsler foli-ationsrdquo Annales Universitatis Mariae Curie-Sklodowska SectioA Mathematica vol 65 no 1 pp 87ndash96 2011
[7] M Abate and G Patrizio Finsler MetricsmdashA Global Approachvol 1591 of Lecture Notes in Mathematics Springer BerlinGermany 1994
[8] R Miron D Hrimiuc H Shimada and S V SabauThe Geom-etry of Hamilton and Lagrange Spaces vol 118 of FundamentalTheories of Physics Kluwer Academic 2001
[9] K Yano ldquoOn a structure defined by a tensor field of type (1 1)satisfying 1198913 + 119891 = 0rdquo Tensor vol 14 pp 99ndash109 1963
[10] M Anastasiei ldquoA framed f-structure on tangent bundle of aFinsler spacerdquoAnalele Universitatii Bucuresti SeriaMatematicavol 49 pp 3ndash9 2000
[11] M Gırtu ldquoAn almost 2-paracontact structure on the cotangentbundle of a Cartan spacerdquoHacettepe Journal ofMathematics andStatistics vol 33 pp 15ndash22 2004
[12] M Gırtu ldquoA Sasakian structure on the indicatrix bundle of aCartan spacerdquo Analele Stiintifice ale Universitatii ldquoAlICuzardquo dinIasi Tomul LIII vol 1 pp 177ndash186 2007
[13] I Mihai R Rosca and L Verstraelen Some Aspects of theDifferential Geometry of Vector Fields vol 2 of Centre forPure and Applied Differential Geometry (PADGE) KatholiekeUniversiteit Leuven Leuven Belgium 1996
[14] A Bejancu and H R Farran ldquoOn the vertical bundle of apseudo-Finsler manifoldrdquo International Journal of Mathematicsand Mathematical Sciences vol 22 no 3 pp 637ndash642 1999
[15] C Ida and A Manea ldquoA vertical Liouville subfoliation on thecotangent bundle of a Cartan space and some related struc-turesrdquo International Journal of Geometric Methods in ModernPhysics vol 11 no 6 21 pages 2014
Submit your manuscripts athttpwwwhindawicom
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Stochastic AnalysisInternational Journal of
2 The Scientific World Journal
that the leaves of F are covering spaces of the leaves ofF andit is called the natural lift ofF to its conormal bundle 119876lowastF
If we denote by 119889119909119886 119886 = 1 119899 the correspondinglocal coframe on119876lowastF thenwe can induce a chart (119909119886 119901
119886 119910119906)
on 119876lowastF where 119901 = 119901119886119889119909119886 isin Γ(119876lowastF) and the system of
equations 119909119886 = const 119901119886= const defines the foliation F
Let 119876F = 119879(119876lowastF)119879F be the normal bundle ofthe foliated manifold (119876lowastF F) The vectors 120597120597119909119886 120597120597119901
119886
119886 = 1 119899 form a natural frame of 119876F at the point(119909119886 119901
119886 119910119906) isin 119876lowastF The canonical projection 120587 119876lowastF rarr
119872 given by 120587(119909119886 119901119886 119910119906) = (119909119886 119910119906) induces another projec-
tion 120587lowast 119879(119876lowastF) rarr 119879119872 which maps the vectors tangent
to F in the vectors tangent toFThus 120587lowastinduces amapping
lowast
119876F rarr 119876F and is denoted by 119881(119876lowastF) = kerlowast
which is a vertical bundle spanned by the vectors 120597120597119901119886
119886 = 1 119899
Lemma 2 Let 119900 119872 rarr 119876lowastF be the zero section of theconormal bundle119876lowastFThen the set 119900(119872) is saturated on119876lowastFwith foliation F
2 Good Vertical Connection on (119876lowastFF)
The purpose of this section is to define a linear connectionnabla X(119881(119876lowastF)) rarr X(Tlowast(119876lowastF) otimes 119881(119876lowastF)) related toconsidered foliated structure where 119876lowastF = 119876lowastF minus 119900(119872)Sincewe have the foliatedmanifold (119876lowastF F) we are lookingfor a Bott connection such that for any vector field119883 tangentto F and any transversal vector field 119884 we have
nabla119883119884 = 119901
119876F ([119883 ]) (2)
where 119901119876F 119879(119876lowastF) rarr 119876F is the canonical projection
and 119901119876F() = 119884
Let us consider now the F-transversal Hamilton-Liou-ville vector field defined by 119862lowast 119876lowastF rarr 119881(119876lowastF)119862lowast(119909119886 119901
119886 119910119906) = 119901
119886(120597120597119901119886) It can be checked that this defini-
tion is well posed From the definition of the Bott connectionthe following lemma holds
Lemma 3 Letnabla X(119881(119876lowastF)) rarr X(119879lowast(119876lowastF)otimes119881(119876lowastF))be a Bott connection Then nabla
119883119862lowast = 0 for every vector field
tangent to F
Now consider the local frame 120597120597119909119886 120597120597119901119886 120597120597119910119906 of
119879(119876lowastF) and recall that the vectors 120597120597119901119886 form the basis of
119881(119876lowastF) With these settings we put
nabla120597120597119909119886
120597
120597119901119887
= Γ119887119886119888
120597
120597119901119888
nabla120597120597119901119886
120597
120597119901119887
= Γ119886119887119888
120597
120597119901119888
nabla120597120597119910119906
120597
120597119901119887
= Γ119887119906119888
120597
120597119901119888
(3)
From the above formulas it follows that
Γ119887119906119888
= 0 nabla120597120597119901119886
119862lowast = (120575119886119888+ 119901119887Γ119886119887119888)
120597
120597119901119888
(4)
The Bott connection nabla allows us to define a mapping
119871 X (119876F) 997888rarr X (119881 (119876lowastF)) 119871 (119883) = nabla119862lowast (5)
where 119901119876F(119883) = 119883 If we denote by Λ the restriction of the
linear mapping 119871 to the bundle 119881(119876lowastF) then we can statethe following
Definition 4 The Bott connection nabla is said to be a goodvertical connection if Λ 119881(119876lowastF) rarr 119881(119876lowastF) is a bundleisomorphism
Observe thatnabla is a good vertical connection if and only ifthematrix 120575119886
119888+119901119887Γ119886119887119888
is nondegenerated If we put119867(119876lowastF) =
ker119871 then we can split the bundle 119876F into direct sum
119876F = 119867 (119876lowastF) oplus 119881 (119876lowastF) (6)
The coefficients of the mapping 119871 in the basis 120597120597119909119886 120597120597119901119886
of 119876F are
119871(120597
120597119909119886) = 119901
119887Γ119887119886119888
120597
120597119901119888
119871 (120597
120597119901119886
) = (120575119886119888+ 119901119887Γ119886119887119888)
120597
120597119901119888
= 119871119886119888
120597
120597119901119888
(7)
It is easy to check that the vectors 120575120575119909119886 = 120597120597119909119886 +
119873119886119887(120597120597119901119887) where119873
119886119887= minus(119871minus1)119888
119887119901119889Γ119889119886119888 form a basis of ker119871
In the sequel we will use the basis 120575120575119909119886 120597120597119901119886 called
adapted as well as its dual 119889119909119886 120575119901119886= 119889119901119886minus 119873119886119887119889119909119887 Using
this coframe we can define the local connection forms by
nabla120597
120597119901119887
= 120596119887119886otimes
120597
120597119901119886
(8)
where
120596119887119886= Γ119887119888119886119889119909119888 + Γ119887119888
119886119889119901119888= (Γ119887119888119886+ Γ119887119889119886119873119889119888) 119889119909119888 + Γ119887119888
119886120575119901119888
= 119867119887119888119886119889119909119888 + Γ119887119888
119886120575119901119888
(9)
Notice that 119867119887119888119886119901119887
= 119873119888119886 The formula 120579(120597120597119901
119886) = 120575120575119909119886
defines a linear mapping 120579 119881(119876lowastF) rarr 119867(119876lowastF)This mapping allows us to extend the connection nabla to thehorizontal bundle119867(119876lowastF) by
nabla119883119884 = 120579 (nabla
119883120579minus1 (119884)) (10)
where 119884 isin Γ(119867(119876lowastF)) 119883 isin Γ(119879(119876lowastF)) In this way weconstruct a linear connection in 119876F
nabla119883119884 = nabla
119883 (V (119884)) + nabla119883 (119884 minus V (119884)) (11)
where 119884 isin Γ(119876F) 119883 isin Γ(119879(119876lowastF)) and V 119876F rarr119881(119876lowastF) is the vertical projection from decomposition (6)In particular we have
nabla120575
120575119909119886= 120596119887119886otimes
120575
120575119909119887 (12)
where 120596119887119886is given in (9)
The Scientific World Journal 3
If 120593 isin Γ(119876lowastF otimes 119876F) is a 1-form with values in 119876Flocally given by
120593 = 120593119886 otimes120575
120575119909119886+ 120593119887otimes
120597
120597119901119887
(13)
then following [1 7] we can define an exterior differential119863120593 by putting
119863120593 = (119889120593119886 minus 120593119886 and 120596119888119886) otimes
120575
120575119909119886+ (119889120593
119887minus 120593119887and 120596119887119888) otimes
120597
120597119901119888
(14)
A straightforward calculus shows that the above formula iswell defined
The bundle 119876lowastF otimes 119876F admits a natural section 120578 givenby
120578 = 119889119909119886 otimes120597
120597119909119886+ 119889119901119887otimes
120597
120597119901119887
= 119889119909119886 otimes120575
120575119909119886+ 120575119901119887otimes
120597
120597119901119887
(15)
It is clear that the form 120578 is well defined
Definition 5 The form 120577 = 119863120578 is called the torsion form ofthe connection nabla
Locally the form 120577 can be expressed as follows
119863120578 = (minus119889119909119886 and 120596119888119886) otimes
120575
120575119909119888+ (119889 (120575119901
119887) minus 120575119901
119888and 120596119888119887) otimes
120597
120597119901119887
= 120577119888 otimes120575
120575119909119888+ 120577119887otimes
120597
120597119901119887
(16)
where
120577119888 =1
2(119867119888119890119886minus 119867119888119886119890) 119889119909119886 and 119889119909119890 minus Γ119888119890
119886119889119909119886 and 120575119901
119890
120577119887= minus119889119873
119888119887and 119889119909119888 minus 119867119886
119890119887120575119901119886and 119889119909119890 minus Γ119886119890
119887120575119901119886and 120575119901119890
(17)
3 Transversal Cartan Metrics on 119876lowastF andRiemannian Foliations
As in the case of transversal Finsler metrics on the normalbundle of a foliation [1 3] a transversal Cartan metric on119876lowastF is a basic function (with respect to the lifted foliationF) 119870 119876lowastF rarr [0infin) which has the following properties
(i) 119870 is 119862infin on 119876lowastF(ii) 119870(119909 120582119901) = 120582119870(119909 119901) for all 120582 gt 0(iii) the 119899 times 119899 matrix (119892119886119887) where 119892119886119887 =
(12)(12059721198702120597119901119886120597119901119886) is positive definite at all
points of 119876lowastFAlso 119870(119909 119901) gt 0 whenever 119901 = 0 As usual [8] the proper-ties of119870 imply that
119901119886 = 119892119886119887119901119887 119901
119886= 119892119886119887119901119887 1198702 = 119892119886119887119901
119886119901119887= 119901119886119901119886
119862119886119887119888119901119888= 119862119886119888119887119901
119888= 119862119888119886119887119901
119888= 0
(18)
where (119892119886119887) is the inverse matrix of (119892119887119886) and we have put
119901119886 = (12)(1205971198702120597119901119886) 119862119886119887119888 = minus(14)(12059731198702120597119901
119886120597119901119887120597119901119888)
Also 119892119886119887 determines a metric structure on 119881(119876lowastF) bysetting
119866V(119883 119884) = 119892119886119887 (119909 119901)119883
119886(119909 119910 119901) 119884
119887(119909 119910 119901) (19)
for every 119883 = 119883119886(119909 119910 119901)(120597120597119901
119886) and 119884 = 119884
119887(119909 119910 119901)(120597
120597119901119887) isin Γ(119881(119876lowastF))Similar reasons as for transversal Finsler foliations (see
Theorem 31 from [1]) lead to the following result
Theorem 6 Let 119870 119876lowastF rarr [0infin) be a transversal Cartanmetric and let 119866V be the Riemannian metric on 119881(119876lowastF)induced by 119870 as in (19) Then there exists exactly one Bottvertical connection nabla X(119881(119876lowastF)) rarr X(119879lowast(119876lowastF) otimes119881(119876lowastF)) such that
(i) nabla is a good vertical connection
(ii) if119883119884 isin Γ(119881(119876lowastF)) and 119885 isin Γ(119879(119876lowastF)) then
119885119866V(119883 119884) = 119866V (nabla
119885119883119884) + 119866V (119883 nabla
119885119884) (20)
(iii) 120577(119883 119884) = 0 for every119883119884 isin Γ(119881(119876lowastF))
(iv) 120577(119883 119884) isin Γ(119881(119876lowastF)) for every119883119884 isin Γ(119867(119876lowastF))
Also the isomorphism 120579 does not depend on the coordi-nates along the leaves of F so the Riemannianmetric in119876F
defined by119866 = 119866ℎ+119866V where119866ℎ(119883 119884) = 119866V(120579minus1(119883) 120579minus1(119884))for every 119883119884 isin Γ(119867(119876lowastF)) and 119866(119883 119884) = 0 for every119883 isin Γ(119867(119876lowastF)) and 119884 isin Γ(119881(119876lowastF)) is a transversalRiemannian metric for the lifted foliation F to the conormalbundle 119876lowastF ofF Hence we can consider the following
Theorem 7 If the conormal bundle of foliationF is endowedwith a transversal Cartan metric then the lifted foliation F tothe conormal bundle 119876lowastF is Riemannian
4 Transversally Framed119891(3120576)-Structures on (119876lowastFF)
The study of structures on manifolds defined by a tensorfield satisfying 1198913 plusmn 119891 = 0 has the origin in a paper byYano [9] Later on these structures have been genericallycalled 119891-structures On the tangent manifold of a Finslerspace the notion of framed119891(3 1)-structure was defined andstudied by Anastasiei in [10] and on the cotangent bundle ofa Cartan space the study is continued in [11 12] Taking intoaccount that the conormal bundle 119876lowastF has a local modelof a cotangent manifold in this section we extend the studyconcerning 119891-structures in our context
Let 120576 = plusmn1 A framed 119891(3 120576)-structure of corank 119904 on a(2119899 + 119904)-dimensional manifold 119873 is a natural generalizationof an almost contact structure on 119873 (for 120576 = 1) and of analmost paracontact structure on119873 (for 120576 = minus1) respectivelyand it is a triplet (119891 (120585
119894) (120596119894)) 119894 = 1 119904 where 119891 is a tensor
4 The Scientific World Journal
field of type (1 1) (120585119894) are vector fields and (120596119894) are 1-forms
on119873 such that
120596119894 (120585119895) = 120575119894119895 119891 (120585
119894) = 0 120596119894 ∘ 119891 = 0
1198912 = minus120576(119868 minussum119894
120596119894 otimes 120585119894)
(21)
where 119868 denotes the Kronecker tensor field on 119873 The nameof 119891(3 120576)-structure was suggested by the identity1198913 +120576119891 = 0For an account of such kind of structures we refer to [13]
Let us consider now that 119876lowastF is endowed with atransversal Cartan metric 119870 The linear operator 120601 given inthe local adapted basis of 119876F by
120601(120575
120575119909119886) = minus120576119892
119886119887
120597
120597119901119887
120601 (120597
120597119901119886
) = 119892119886119887120575
120575119909119887(22)
defines an almost complex structure on 119876F for 120576 = 1 and analmost paracomplex structure on119876F for 120576 = minus1 respectivelyWe also have
119866 (120601 (119883) 120601 (119884)) = 119866 (119883 119884) forall119883 119884 isin Γ (119876F) (23)
Let us put 1205851= (119901119886119870)(120575120575119909119886) and 120585
2= (119901119886119870)(120597120597119901
119886) =
(1119870)119862lowast Thus we have two global transverse vector fieldson (119876lowastF F) which are linearly independent The first istransversally horizontal and the second one is transversallyvertical
From the definition of 120601 it follows
120601 (1205851) = minus120576120585
2 120601 (120585
2) = 1205851 (24)
Now if we consider the dual transverse 1-forms of 1205851and
1205852 respectively locally given by 1205961 = (119901
119886119870)119889119909119886 and 1205962 =
(119901119886119870)120575119901119886 then we easily check that
1205961 ∘ 120601 = 1205962 1205962 ∘ 120601 = minus1205761205961
1205961 (119883) = 119866 (119883 1205851) 1205962 (119883) = 119866 (119883 120585
2) forall119883 isin Γ (119876F)
(25)
Next using120601 120585119894 and120596119894 119894 isin 1 2 we construct the transverse
tensor field 119891 of type (1 1) on (119876lowastF F) by putting
119891 (119883) = 120601 (119883) minus 1205962 (119883) 1205851 + 1205761205961 (119883) 1205852 119883 isin Γ (119876F)
(26)
Using a similar argument as in [6 10ndash12] by direct calculuswe obtain the following
Theorem 8 The triple (119891 (120585119894) (120596119894)) 119894 isin 1 2 provides
some transversally framed 119891(3 120576)-structures of corank 2 on(119876lowastF F) that is the following hold
(i) 120596119894(120585119895) = 120575119894119895 119891(120585119894) = 0 120596119894 ∘ 119891 = 0
(ii) 1198912 = minus120576(119868 minus 1205961 otimes 1205851minus 1205962 otimes 120585
2)
(iii) 119891 is of rank 2119899 minus 2 and 1198913 + 120576119891 = 0
Theorem 9 The transversal Riemannian metric 119866 on(119876lowastF F) satisfies
119866 (119891 (119883) 119891 (119884)) = 119866 (119883 119884) minus 1205961 (119883) 1205961(119884)
minus 1205962 (119883) 1205962(119884) 119883 119884 isin Γ (119876F)
(27)
For 120576 = 1 we put
Φ (119883 119884) = 119866 (119891 (119883) 119884) 119883 119884 isin Γ (119876F) (28)
By usingTheorems 8 and 9 we obtain
Φ (119883 119884) = minusΦ (119884119883) 119883 119884 isin Γ (119876F) (29)
ThusΦ is a transverse 2-form on (119876lowastF F) It is degeneratewith null space span120585
1 1205852
Also using the calculus in local coordinates we easilyobtain
Φ(120575
120575119909119886
120575
120575119909119887) = 0 Φ(
120575
120575119909119886
120597
120597119901119887
) = minus120575119887119886+119901119886119901119887
1198702
Φ (120597
120597119901119886
120597
120597119901119887
) = 0
(30)
On the other hand we have
1198891205961 (120575
120575119909119886
120575
120575119909119887) =
120575
120575119909119886(119901119887
119870) minus
120575
120575119909119887(119901119886
119870)
1198891205961 (120575
120575119909119886
120597
120597119901119887
) =1
119870(minus120575119887119886+119901119886119901119887
1198702)
1198891205961 (120597
120597119901119886
120597
120597119901119887
) = 0
(31)
where in the second relation we have used 119901119886 = 119870(120597119870120597119901119886)
which follows from the homogeneity conditions (18) of 119870Comparing now Φ with 1198891205961 we obtain
1
119870Φ = 1198891205961 + Ψ (32)
where Ψ = ((120575120575119909119887)(119901119886119870) minus (120575120575119909119886)(119901
119887119870))119889119909119886 and 119889119909119887
Thus (1119870)Φ is transversally closed if and only if Ψ istransversally closed Concluding (1119870)Φ is in general analmost transversally presymplectic structure on (119876lowastF F)
Similarly for 120576 = minus1 we can put
119867(119883 119884) = 119866 (119891 (119883) 119884) 119883 119884 isin Γ (119876F) (33)
We have the following
Theorem 10 The mapping119867 is a symmetric bilinear form on(119876lowastF F) and the annihilator of119867 is ker119891
The Scientific World Journal 5
Proof The symmetry and bilinearity are obvious Also thenull space of119867 is
119883 isin Γ (119876F) | 119867 (119883 119884) = 0 forall119884 isin Γ (119876F)
= 119883 isin Γ (119876F) | 119866 (119891 (119883) 119884) = 0 = ker119891(34)
which end the proof
Locally we obtain
119867 = (119892119886119887
minus119901119886119901119887
1198702)119889119909119886 otimes 119889119909119887 minus (119892119886119887 minus
119901119886119901119887
1198702)120575119901119886otimes 120575119901119887
(35)
with det(119892119886119887
minus (1199011198861199011198871198702)) = 0 since (119892
119886119887minus (1199011198861199011198871198702))119901119887 =
119901119886minus 119901119886= 0 and similarly det(119892119886119887 minus (1199011198861199011198871198702)) = 0 since
(119892119886119887 minus (1199011198861199011198871198702))119901119887= 119901119886 minus 119901119886 = 0
Remark 11 The map 119867 is a transversally singular pseudo-Riemannian metric on (119876lowastF F)
5 An Almost (Para)Contact Structure onTransverse Liouville Distribution of(119876lowastFF)
Denote by 1205852 the line vector bundle over119876lowastF spanned by 120585
2
and we define the transverse vertical Liouville distribution asthe complementary orthogonal distribution 119878(119876lowastF) to 120585
2
in119881(119876lowastF)with respect to119866V namely119881(119876lowastF) = 119878(119876lowastF)oplus
1205852 Hence 119878(119876lowastF) is defined by 120572 = 1205962|
119881(119876lowastF) that is
Γ (119878 (119876lowastF)) = 119883 isin Γ (119881 (119876lowastF)) 120572 (119883) = 0 (36)
Thus any transverse vertical vector field119883 isin Γ(119881(119876lowastF)) canbe expressed as
119883 = 119875119883 + 120572 (119883) 1205852 (37)
where 119875 is the projection morphism of 119881(119876lowastF) on 119878(119876lowastF)By direct calculations one gets the following
Proposition 12 For any transverse vertical vector fields119883119884 isinΓ(119881(119876lowastF)) one has
119866V(119883 119875119884) = 119866V
(119875119883 119875119884) = 119866V(119883 119884) minus 120572 (119883) 120572 (119884) (38)
Theorem 13 The transverse vertical Liouville distribution119878(119876lowastF) is integrable
Proof The proof follows using an argument similar to Theo-rem 31 [14] (see alsoTheorem 21 [6] orTheorem 4 [15])
In the following we will consider the transverse Liouvilledistribution of (119876lowastF F) as the complementary orthogonaldistribution 119878(119876lowastF) to 120585
2 in 119876F with respect to 119866 that is
119878(119876lowastF) = 119867(119876lowastF) oplus 119878(119876lowastF)Let us restrict to 119878(119876lowastF) all the geometrical structures
introduced in Section 4 for all 119876F We indicate this byoverlines Hence we have
(i) 1205851= 1205851since 120585
1lies in 119878(119876lowastF)
(ii) 1205962 = 0 since 1205962(119883) = 119866(119883 1205852) = 0 for every
transverse vector field119883 isin 119878(119876lowastF)(iii) 119866 = 119866|
119878(119876lowastF)
(iv) 119891(119883) = 120601(119883) + 1205761205961(119883) otimes 1205852is an endomorphism of
119878(119876lowastF) since
119866(119891 (119883) 1205852) = 119866 (120601 (119883) 1205852) + 1205761205961 (119883)119866 (1205852 1205852)
= 1205962 (120601 (119883)) + 1205761205961 (119883) = 0(39)
We denote now 120585 = 1205851and 120578 = 1205961
By Theorem 8 we obtain the following
Theorem 14 The triple (119891 120585 120578) provides an almost(para)contact structure on 119878(119876lowastF) that is
(i) 1198913
+ 120576119891 = 0 119903119886119899119896119891 = 2119899 minus 2 = (2119899 minus 1) minus 1
(ii) 120578(120585) = 1 119891(120585) = 0 120578 ∘ 119891 = 0
(iii) 1198912
(119883) = minus120576(119883 minus 120578(119883)120585) for119883 isin 119878(119876lowastF)
Also by Theorem 9 we obtain the following
Theorem 15 The transversal Riemannian metric 119866 verifies
119866(119891 (119883) 119891 (119884)) = 119866 (119883 119884) minus 120578 (119883) 120578 (119884) (40)
for every transverse vector fields119883119884 isin 119878(119876lowastF)
Concluding the ensemble (119891 120585 120578 119866) is an almost(para)contact Riemannian structure on 119878(119876lowastF)
Conflict of Interests
The authors declare that they have no conflict of interests
Authorsrsquo Contribution
Both authors contributed equally to the paper All the authorsread and approved the final paper
Acknowledgment
The first author is supported by the Grant ldquoFellowship ofTransilvania University of Brasov 2014rdquo
References
[1] A Miernowski andWMozgawa ldquoLift of the Finsler foliation toits normal bundlerdquo Differential Geometry and Its Applicationsvol 24 no 2 pp 209ndash214 2006
[2] P Popescu and C Ida ldquoFoliations compatible with hamilto-niansrdquo in Proceedings of the BSG International Conference onDifferential Geometry and Dynamical Systems (DGDS rsquo13) pp147ndash155 Bucharest Romania October 2013
6 The Scientific World Journal
[3] P Popescu and M Popescu ldquoLagrangians adapted to submer-sions and foliationsrdquoDifferential Geometry and Its Applicationsvol 27 no 2 pp 171ndash178 2009
[4] P Popescu and M Popescu ldquoFoliated vector bundles andRiemannian foliationsrdquo Comptes Rendus Mathematique vol349 no 7-8 pp 445ndash449 2011
[5] P Popescu and M Popescu ldquoLagrangians and Hamiltonianstructures and foliationsrdquo Physics AUC vol 21 no 1 pp 203ndash206 2011
[6] C Ida ldquoSome framed 119891-structures on transversally Finsler foli-ationsrdquo Annales Universitatis Mariae Curie-Sklodowska SectioA Mathematica vol 65 no 1 pp 87ndash96 2011
[7] M Abate and G Patrizio Finsler MetricsmdashA Global Approachvol 1591 of Lecture Notes in Mathematics Springer BerlinGermany 1994
[8] R Miron D Hrimiuc H Shimada and S V SabauThe Geom-etry of Hamilton and Lagrange Spaces vol 118 of FundamentalTheories of Physics Kluwer Academic 2001
[9] K Yano ldquoOn a structure defined by a tensor field of type (1 1)satisfying 1198913 + 119891 = 0rdquo Tensor vol 14 pp 99ndash109 1963
[10] M Anastasiei ldquoA framed f-structure on tangent bundle of aFinsler spacerdquoAnalele Universitatii Bucuresti SeriaMatematicavol 49 pp 3ndash9 2000
[11] M Gırtu ldquoAn almost 2-paracontact structure on the cotangentbundle of a Cartan spacerdquoHacettepe Journal ofMathematics andStatistics vol 33 pp 15ndash22 2004
[12] M Gırtu ldquoA Sasakian structure on the indicatrix bundle of aCartan spacerdquo Analele Stiintifice ale Universitatii ldquoAlICuzardquo dinIasi Tomul LIII vol 1 pp 177ndash186 2007
[13] I Mihai R Rosca and L Verstraelen Some Aspects of theDifferential Geometry of Vector Fields vol 2 of Centre forPure and Applied Differential Geometry (PADGE) KatholiekeUniversiteit Leuven Leuven Belgium 1996
[14] A Bejancu and H R Farran ldquoOn the vertical bundle of apseudo-Finsler manifoldrdquo International Journal of Mathematicsand Mathematical Sciences vol 22 no 3 pp 637ndash642 1999
[15] C Ida and A Manea ldquoA vertical Liouville subfoliation on thecotangent bundle of a Cartan space and some related struc-turesrdquo International Journal of Geometric Methods in ModernPhysics vol 11 no 6 21 pages 2014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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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
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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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
The Scientific World Journal 3
If 120593 isin Γ(119876lowastF otimes 119876F) is a 1-form with values in 119876Flocally given by
120593 = 120593119886 otimes120575
120575119909119886+ 120593119887otimes
120597
120597119901119887
(13)
then following [1 7] we can define an exterior differential119863120593 by putting
119863120593 = (119889120593119886 minus 120593119886 and 120596119888119886) otimes
120575
120575119909119886+ (119889120593
119887minus 120593119887and 120596119887119888) otimes
120597
120597119901119888
(14)
A straightforward calculus shows that the above formula iswell defined
The bundle 119876lowastF otimes 119876F admits a natural section 120578 givenby
120578 = 119889119909119886 otimes120597
120597119909119886+ 119889119901119887otimes
120597
120597119901119887
= 119889119909119886 otimes120575
120575119909119886+ 120575119901119887otimes
120597
120597119901119887
(15)
It is clear that the form 120578 is well defined
Definition 5 The form 120577 = 119863120578 is called the torsion form ofthe connection nabla
Locally the form 120577 can be expressed as follows
119863120578 = (minus119889119909119886 and 120596119888119886) otimes
120575
120575119909119888+ (119889 (120575119901
119887) minus 120575119901
119888and 120596119888119887) otimes
120597
120597119901119887
= 120577119888 otimes120575
120575119909119888+ 120577119887otimes
120597
120597119901119887
(16)
where
120577119888 =1
2(119867119888119890119886minus 119867119888119886119890) 119889119909119886 and 119889119909119890 minus Γ119888119890
119886119889119909119886 and 120575119901
119890
120577119887= minus119889119873
119888119887and 119889119909119888 minus 119867119886
119890119887120575119901119886and 119889119909119890 minus Γ119886119890
119887120575119901119886and 120575119901119890
(17)
3 Transversal Cartan Metrics on 119876lowastF andRiemannian Foliations
As in the case of transversal Finsler metrics on the normalbundle of a foliation [1 3] a transversal Cartan metric on119876lowastF is a basic function (with respect to the lifted foliationF) 119870 119876lowastF rarr [0infin) which has the following properties
(i) 119870 is 119862infin on 119876lowastF(ii) 119870(119909 120582119901) = 120582119870(119909 119901) for all 120582 gt 0(iii) the 119899 times 119899 matrix (119892119886119887) where 119892119886119887 =
(12)(12059721198702120597119901119886120597119901119886) is positive definite at all
points of 119876lowastFAlso 119870(119909 119901) gt 0 whenever 119901 = 0 As usual [8] the proper-ties of119870 imply that
119901119886 = 119892119886119887119901119887 119901
119886= 119892119886119887119901119887 1198702 = 119892119886119887119901
119886119901119887= 119901119886119901119886
119862119886119887119888119901119888= 119862119886119888119887119901
119888= 119862119888119886119887119901
119888= 0
(18)
where (119892119886119887) is the inverse matrix of (119892119887119886) and we have put
119901119886 = (12)(1205971198702120597119901119886) 119862119886119887119888 = minus(14)(12059731198702120597119901
119886120597119901119887120597119901119888)
Also 119892119886119887 determines a metric structure on 119881(119876lowastF) bysetting
119866V(119883 119884) = 119892119886119887 (119909 119901)119883
119886(119909 119910 119901) 119884
119887(119909 119910 119901) (19)
for every 119883 = 119883119886(119909 119910 119901)(120597120597119901
119886) and 119884 = 119884
119887(119909 119910 119901)(120597
120597119901119887) isin Γ(119881(119876lowastF))Similar reasons as for transversal Finsler foliations (see
Theorem 31 from [1]) lead to the following result
Theorem 6 Let 119870 119876lowastF rarr [0infin) be a transversal Cartanmetric and let 119866V be the Riemannian metric on 119881(119876lowastF)induced by 119870 as in (19) Then there exists exactly one Bottvertical connection nabla X(119881(119876lowastF)) rarr X(119879lowast(119876lowastF) otimes119881(119876lowastF)) such that
(i) nabla is a good vertical connection
(ii) if119883119884 isin Γ(119881(119876lowastF)) and 119885 isin Γ(119879(119876lowastF)) then
119885119866V(119883 119884) = 119866V (nabla
119885119883119884) + 119866V (119883 nabla
119885119884) (20)
(iii) 120577(119883 119884) = 0 for every119883119884 isin Γ(119881(119876lowastF))
(iv) 120577(119883 119884) isin Γ(119881(119876lowastF)) for every119883119884 isin Γ(119867(119876lowastF))
Also the isomorphism 120579 does not depend on the coordi-nates along the leaves of F so the Riemannianmetric in119876F
defined by119866 = 119866ℎ+119866V where119866ℎ(119883 119884) = 119866V(120579minus1(119883) 120579minus1(119884))for every 119883119884 isin Γ(119867(119876lowastF)) and 119866(119883 119884) = 0 for every119883 isin Γ(119867(119876lowastF)) and 119884 isin Γ(119881(119876lowastF)) is a transversalRiemannian metric for the lifted foliation F to the conormalbundle 119876lowastF ofF Hence we can consider the following
Theorem 7 If the conormal bundle of foliationF is endowedwith a transversal Cartan metric then the lifted foliation F tothe conormal bundle 119876lowastF is Riemannian
4 Transversally Framed119891(3120576)-Structures on (119876lowastFF)
The study of structures on manifolds defined by a tensorfield satisfying 1198913 plusmn 119891 = 0 has the origin in a paper byYano [9] Later on these structures have been genericallycalled 119891-structures On the tangent manifold of a Finslerspace the notion of framed119891(3 1)-structure was defined andstudied by Anastasiei in [10] and on the cotangent bundle ofa Cartan space the study is continued in [11 12] Taking intoaccount that the conormal bundle 119876lowastF has a local modelof a cotangent manifold in this section we extend the studyconcerning 119891-structures in our context
Let 120576 = plusmn1 A framed 119891(3 120576)-structure of corank 119904 on a(2119899 + 119904)-dimensional manifold 119873 is a natural generalizationof an almost contact structure on 119873 (for 120576 = 1) and of analmost paracontact structure on119873 (for 120576 = minus1) respectivelyand it is a triplet (119891 (120585
119894) (120596119894)) 119894 = 1 119904 where 119891 is a tensor
4 The Scientific World Journal
field of type (1 1) (120585119894) are vector fields and (120596119894) are 1-forms
on119873 such that
120596119894 (120585119895) = 120575119894119895 119891 (120585
119894) = 0 120596119894 ∘ 119891 = 0
1198912 = minus120576(119868 minussum119894
120596119894 otimes 120585119894)
(21)
where 119868 denotes the Kronecker tensor field on 119873 The nameof 119891(3 120576)-structure was suggested by the identity1198913 +120576119891 = 0For an account of such kind of structures we refer to [13]
Let us consider now that 119876lowastF is endowed with atransversal Cartan metric 119870 The linear operator 120601 given inthe local adapted basis of 119876F by
120601(120575
120575119909119886) = minus120576119892
119886119887
120597
120597119901119887
120601 (120597
120597119901119886
) = 119892119886119887120575
120575119909119887(22)
defines an almost complex structure on 119876F for 120576 = 1 and analmost paracomplex structure on119876F for 120576 = minus1 respectivelyWe also have
119866 (120601 (119883) 120601 (119884)) = 119866 (119883 119884) forall119883 119884 isin Γ (119876F) (23)
Let us put 1205851= (119901119886119870)(120575120575119909119886) and 120585
2= (119901119886119870)(120597120597119901
119886) =
(1119870)119862lowast Thus we have two global transverse vector fieldson (119876lowastF F) which are linearly independent The first istransversally horizontal and the second one is transversallyvertical
From the definition of 120601 it follows
120601 (1205851) = minus120576120585
2 120601 (120585
2) = 1205851 (24)
Now if we consider the dual transverse 1-forms of 1205851and
1205852 respectively locally given by 1205961 = (119901
119886119870)119889119909119886 and 1205962 =
(119901119886119870)120575119901119886 then we easily check that
1205961 ∘ 120601 = 1205962 1205962 ∘ 120601 = minus1205761205961
1205961 (119883) = 119866 (119883 1205851) 1205962 (119883) = 119866 (119883 120585
2) forall119883 isin Γ (119876F)
(25)
Next using120601 120585119894 and120596119894 119894 isin 1 2 we construct the transverse
tensor field 119891 of type (1 1) on (119876lowastF F) by putting
119891 (119883) = 120601 (119883) minus 1205962 (119883) 1205851 + 1205761205961 (119883) 1205852 119883 isin Γ (119876F)
(26)
Using a similar argument as in [6 10ndash12] by direct calculuswe obtain the following
Theorem 8 The triple (119891 (120585119894) (120596119894)) 119894 isin 1 2 provides
some transversally framed 119891(3 120576)-structures of corank 2 on(119876lowastF F) that is the following hold
(i) 120596119894(120585119895) = 120575119894119895 119891(120585119894) = 0 120596119894 ∘ 119891 = 0
(ii) 1198912 = minus120576(119868 minus 1205961 otimes 1205851minus 1205962 otimes 120585
2)
(iii) 119891 is of rank 2119899 minus 2 and 1198913 + 120576119891 = 0
Theorem 9 The transversal Riemannian metric 119866 on(119876lowastF F) satisfies
119866 (119891 (119883) 119891 (119884)) = 119866 (119883 119884) minus 1205961 (119883) 1205961(119884)
minus 1205962 (119883) 1205962(119884) 119883 119884 isin Γ (119876F)
(27)
For 120576 = 1 we put
Φ (119883 119884) = 119866 (119891 (119883) 119884) 119883 119884 isin Γ (119876F) (28)
By usingTheorems 8 and 9 we obtain
Φ (119883 119884) = minusΦ (119884119883) 119883 119884 isin Γ (119876F) (29)
ThusΦ is a transverse 2-form on (119876lowastF F) It is degeneratewith null space span120585
1 1205852
Also using the calculus in local coordinates we easilyobtain
Φ(120575
120575119909119886
120575
120575119909119887) = 0 Φ(
120575
120575119909119886
120597
120597119901119887
) = minus120575119887119886+119901119886119901119887
1198702
Φ (120597
120597119901119886
120597
120597119901119887
) = 0
(30)
On the other hand we have
1198891205961 (120575
120575119909119886
120575
120575119909119887) =
120575
120575119909119886(119901119887
119870) minus
120575
120575119909119887(119901119886
119870)
1198891205961 (120575
120575119909119886
120597
120597119901119887
) =1
119870(minus120575119887119886+119901119886119901119887
1198702)
1198891205961 (120597
120597119901119886
120597
120597119901119887
) = 0
(31)
where in the second relation we have used 119901119886 = 119870(120597119870120597119901119886)
which follows from the homogeneity conditions (18) of 119870Comparing now Φ with 1198891205961 we obtain
1
119870Φ = 1198891205961 + Ψ (32)
where Ψ = ((120575120575119909119887)(119901119886119870) minus (120575120575119909119886)(119901
119887119870))119889119909119886 and 119889119909119887
Thus (1119870)Φ is transversally closed if and only if Ψ istransversally closed Concluding (1119870)Φ is in general analmost transversally presymplectic structure on (119876lowastF F)
Similarly for 120576 = minus1 we can put
119867(119883 119884) = 119866 (119891 (119883) 119884) 119883 119884 isin Γ (119876F) (33)
We have the following
Theorem 10 The mapping119867 is a symmetric bilinear form on(119876lowastF F) and the annihilator of119867 is ker119891
The Scientific World Journal 5
Proof The symmetry and bilinearity are obvious Also thenull space of119867 is
119883 isin Γ (119876F) | 119867 (119883 119884) = 0 forall119884 isin Γ (119876F)
= 119883 isin Γ (119876F) | 119866 (119891 (119883) 119884) = 0 = ker119891(34)
which end the proof
Locally we obtain
119867 = (119892119886119887
minus119901119886119901119887
1198702)119889119909119886 otimes 119889119909119887 minus (119892119886119887 minus
119901119886119901119887
1198702)120575119901119886otimes 120575119901119887
(35)
with det(119892119886119887
minus (1199011198861199011198871198702)) = 0 since (119892
119886119887minus (1199011198861199011198871198702))119901119887 =
119901119886minus 119901119886= 0 and similarly det(119892119886119887 minus (1199011198861199011198871198702)) = 0 since
(119892119886119887 minus (1199011198861199011198871198702))119901119887= 119901119886 minus 119901119886 = 0
Remark 11 The map 119867 is a transversally singular pseudo-Riemannian metric on (119876lowastF F)
5 An Almost (Para)Contact Structure onTransverse Liouville Distribution of(119876lowastFF)
Denote by 1205852 the line vector bundle over119876lowastF spanned by 120585
2
and we define the transverse vertical Liouville distribution asthe complementary orthogonal distribution 119878(119876lowastF) to 120585
2
in119881(119876lowastF)with respect to119866V namely119881(119876lowastF) = 119878(119876lowastF)oplus
1205852 Hence 119878(119876lowastF) is defined by 120572 = 1205962|
119881(119876lowastF) that is
Γ (119878 (119876lowastF)) = 119883 isin Γ (119881 (119876lowastF)) 120572 (119883) = 0 (36)
Thus any transverse vertical vector field119883 isin Γ(119881(119876lowastF)) canbe expressed as
119883 = 119875119883 + 120572 (119883) 1205852 (37)
where 119875 is the projection morphism of 119881(119876lowastF) on 119878(119876lowastF)By direct calculations one gets the following
Proposition 12 For any transverse vertical vector fields119883119884 isinΓ(119881(119876lowastF)) one has
119866V(119883 119875119884) = 119866V
(119875119883 119875119884) = 119866V(119883 119884) minus 120572 (119883) 120572 (119884) (38)
Theorem 13 The transverse vertical Liouville distribution119878(119876lowastF) is integrable
Proof The proof follows using an argument similar to Theo-rem 31 [14] (see alsoTheorem 21 [6] orTheorem 4 [15])
In the following we will consider the transverse Liouvilledistribution of (119876lowastF F) as the complementary orthogonaldistribution 119878(119876lowastF) to 120585
2 in 119876F with respect to 119866 that is
119878(119876lowastF) = 119867(119876lowastF) oplus 119878(119876lowastF)Let us restrict to 119878(119876lowastF) all the geometrical structures
introduced in Section 4 for all 119876F We indicate this byoverlines Hence we have
(i) 1205851= 1205851since 120585
1lies in 119878(119876lowastF)
(ii) 1205962 = 0 since 1205962(119883) = 119866(119883 1205852) = 0 for every
transverse vector field119883 isin 119878(119876lowastF)(iii) 119866 = 119866|
119878(119876lowastF)
(iv) 119891(119883) = 120601(119883) + 1205761205961(119883) otimes 1205852is an endomorphism of
119878(119876lowastF) since
119866(119891 (119883) 1205852) = 119866 (120601 (119883) 1205852) + 1205761205961 (119883)119866 (1205852 1205852)
= 1205962 (120601 (119883)) + 1205761205961 (119883) = 0(39)
We denote now 120585 = 1205851and 120578 = 1205961
By Theorem 8 we obtain the following
Theorem 14 The triple (119891 120585 120578) provides an almost(para)contact structure on 119878(119876lowastF) that is
(i) 1198913
+ 120576119891 = 0 119903119886119899119896119891 = 2119899 minus 2 = (2119899 minus 1) minus 1
(ii) 120578(120585) = 1 119891(120585) = 0 120578 ∘ 119891 = 0
(iii) 1198912
(119883) = minus120576(119883 minus 120578(119883)120585) for119883 isin 119878(119876lowastF)
Also by Theorem 9 we obtain the following
Theorem 15 The transversal Riemannian metric 119866 verifies
119866(119891 (119883) 119891 (119884)) = 119866 (119883 119884) minus 120578 (119883) 120578 (119884) (40)
for every transverse vector fields119883119884 isin 119878(119876lowastF)
Concluding the ensemble (119891 120585 120578 119866) is an almost(para)contact Riemannian structure on 119878(119876lowastF)
Conflict of Interests
The authors declare that they have no conflict of interests
Authorsrsquo Contribution
Both authors contributed equally to the paper All the authorsread and approved the final paper
Acknowledgment
The first author is supported by the Grant ldquoFellowship ofTransilvania University of Brasov 2014rdquo
References
[1] A Miernowski andWMozgawa ldquoLift of the Finsler foliation toits normal bundlerdquo Differential Geometry and Its Applicationsvol 24 no 2 pp 209ndash214 2006
[2] P Popescu and C Ida ldquoFoliations compatible with hamilto-niansrdquo in Proceedings of the BSG International Conference onDifferential Geometry and Dynamical Systems (DGDS rsquo13) pp147ndash155 Bucharest Romania October 2013
6 The Scientific World Journal
[3] P Popescu and M Popescu ldquoLagrangians adapted to submer-sions and foliationsrdquoDifferential Geometry and Its Applicationsvol 27 no 2 pp 171ndash178 2009
[4] P Popescu and M Popescu ldquoFoliated vector bundles andRiemannian foliationsrdquo Comptes Rendus Mathematique vol349 no 7-8 pp 445ndash449 2011
[5] P Popescu and M Popescu ldquoLagrangians and Hamiltonianstructures and foliationsrdquo Physics AUC vol 21 no 1 pp 203ndash206 2011
[6] C Ida ldquoSome framed 119891-structures on transversally Finsler foli-ationsrdquo Annales Universitatis Mariae Curie-Sklodowska SectioA Mathematica vol 65 no 1 pp 87ndash96 2011
[7] M Abate and G Patrizio Finsler MetricsmdashA Global Approachvol 1591 of Lecture Notes in Mathematics Springer BerlinGermany 1994
[8] R Miron D Hrimiuc H Shimada and S V SabauThe Geom-etry of Hamilton and Lagrange Spaces vol 118 of FundamentalTheories of Physics Kluwer Academic 2001
[9] K Yano ldquoOn a structure defined by a tensor field of type (1 1)satisfying 1198913 + 119891 = 0rdquo Tensor vol 14 pp 99ndash109 1963
[10] M Anastasiei ldquoA framed f-structure on tangent bundle of aFinsler spacerdquoAnalele Universitatii Bucuresti SeriaMatematicavol 49 pp 3ndash9 2000
[11] M Gırtu ldquoAn almost 2-paracontact structure on the cotangentbundle of a Cartan spacerdquoHacettepe Journal ofMathematics andStatistics vol 33 pp 15ndash22 2004
[12] M Gırtu ldquoA Sasakian structure on the indicatrix bundle of aCartan spacerdquo Analele Stiintifice ale Universitatii ldquoAlICuzardquo dinIasi Tomul LIII vol 1 pp 177ndash186 2007
[13] I Mihai R Rosca and L Verstraelen Some Aspects of theDifferential Geometry of Vector Fields vol 2 of Centre forPure and Applied Differential Geometry (PADGE) KatholiekeUniversiteit Leuven Leuven Belgium 1996
[14] A Bejancu and H R Farran ldquoOn the vertical bundle of apseudo-Finsler manifoldrdquo International Journal of Mathematicsand Mathematical Sciences vol 22 no 3 pp 637ndash642 1999
[15] C Ida and A Manea ldquoA vertical Liouville subfoliation on thecotangent bundle of a Cartan space and some related struc-turesrdquo International Journal of Geometric Methods in ModernPhysics vol 11 no 6 21 pages 2014
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
4 The Scientific World Journal
field of type (1 1) (120585119894) are vector fields and (120596119894) are 1-forms
on119873 such that
120596119894 (120585119895) = 120575119894119895 119891 (120585
119894) = 0 120596119894 ∘ 119891 = 0
1198912 = minus120576(119868 minussum119894
120596119894 otimes 120585119894)
(21)
where 119868 denotes the Kronecker tensor field on 119873 The nameof 119891(3 120576)-structure was suggested by the identity1198913 +120576119891 = 0For an account of such kind of structures we refer to [13]
Let us consider now that 119876lowastF is endowed with atransversal Cartan metric 119870 The linear operator 120601 given inthe local adapted basis of 119876F by
120601(120575
120575119909119886) = minus120576119892
119886119887
120597
120597119901119887
120601 (120597
120597119901119886
) = 119892119886119887120575
120575119909119887(22)
defines an almost complex structure on 119876F for 120576 = 1 and analmost paracomplex structure on119876F for 120576 = minus1 respectivelyWe also have
119866 (120601 (119883) 120601 (119884)) = 119866 (119883 119884) forall119883 119884 isin Γ (119876F) (23)
Let us put 1205851= (119901119886119870)(120575120575119909119886) and 120585
2= (119901119886119870)(120597120597119901
119886) =
(1119870)119862lowast Thus we have two global transverse vector fieldson (119876lowastF F) which are linearly independent The first istransversally horizontal and the second one is transversallyvertical
From the definition of 120601 it follows
120601 (1205851) = minus120576120585
2 120601 (120585
2) = 1205851 (24)
Now if we consider the dual transverse 1-forms of 1205851and
1205852 respectively locally given by 1205961 = (119901
119886119870)119889119909119886 and 1205962 =
(119901119886119870)120575119901119886 then we easily check that
1205961 ∘ 120601 = 1205962 1205962 ∘ 120601 = minus1205761205961
1205961 (119883) = 119866 (119883 1205851) 1205962 (119883) = 119866 (119883 120585
2) forall119883 isin Γ (119876F)
(25)
Next using120601 120585119894 and120596119894 119894 isin 1 2 we construct the transverse
tensor field 119891 of type (1 1) on (119876lowastF F) by putting
119891 (119883) = 120601 (119883) minus 1205962 (119883) 1205851 + 1205761205961 (119883) 1205852 119883 isin Γ (119876F)
(26)
Using a similar argument as in [6 10ndash12] by direct calculuswe obtain the following
Theorem 8 The triple (119891 (120585119894) (120596119894)) 119894 isin 1 2 provides
some transversally framed 119891(3 120576)-structures of corank 2 on(119876lowastF F) that is the following hold
(i) 120596119894(120585119895) = 120575119894119895 119891(120585119894) = 0 120596119894 ∘ 119891 = 0
(ii) 1198912 = minus120576(119868 minus 1205961 otimes 1205851minus 1205962 otimes 120585
2)
(iii) 119891 is of rank 2119899 minus 2 and 1198913 + 120576119891 = 0
Theorem 9 The transversal Riemannian metric 119866 on(119876lowastF F) satisfies
119866 (119891 (119883) 119891 (119884)) = 119866 (119883 119884) minus 1205961 (119883) 1205961(119884)
minus 1205962 (119883) 1205962(119884) 119883 119884 isin Γ (119876F)
(27)
For 120576 = 1 we put
Φ (119883 119884) = 119866 (119891 (119883) 119884) 119883 119884 isin Γ (119876F) (28)
By usingTheorems 8 and 9 we obtain
Φ (119883 119884) = minusΦ (119884119883) 119883 119884 isin Γ (119876F) (29)
ThusΦ is a transverse 2-form on (119876lowastF F) It is degeneratewith null space span120585
1 1205852
Also using the calculus in local coordinates we easilyobtain
Φ(120575
120575119909119886
120575
120575119909119887) = 0 Φ(
120575
120575119909119886
120597
120597119901119887
) = minus120575119887119886+119901119886119901119887
1198702
Φ (120597
120597119901119886
120597
120597119901119887
) = 0
(30)
On the other hand we have
1198891205961 (120575
120575119909119886
120575
120575119909119887) =
120575
120575119909119886(119901119887
119870) minus
120575
120575119909119887(119901119886
119870)
1198891205961 (120575
120575119909119886
120597
120597119901119887
) =1
119870(minus120575119887119886+119901119886119901119887
1198702)
1198891205961 (120597
120597119901119886
120597
120597119901119887
) = 0
(31)
where in the second relation we have used 119901119886 = 119870(120597119870120597119901119886)
which follows from the homogeneity conditions (18) of 119870Comparing now Φ with 1198891205961 we obtain
1
119870Φ = 1198891205961 + Ψ (32)
where Ψ = ((120575120575119909119887)(119901119886119870) minus (120575120575119909119886)(119901
119887119870))119889119909119886 and 119889119909119887
Thus (1119870)Φ is transversally closed if and only if Ψ istransversally closed Concluding (1119870)Φ is in general analmost transversally presymplectic structure on (119876lowastF F)
Similarly for 120576 = minus1 we can put
119867(119883 119884) = 119866 (119891 (119883) 119884) 119883 119884 isin Γ (119876F) (33)
We have the following
Theorem 10 The mapping119867 is a symmetric bilinear form on(119876lowastF F) and the annihilator of119867 is ker119891
The Scientific World Journal 5
Proof The symmetry and bilinearity are obvious Also thenull space of119867 is
119883 isin Γ (119876F) | 119867 (119883 119884) = 0 forall119884 isin Γ (119876F)
= 119883 isin Γ (119876F) | 119866 (119891 (119883) 119884) = 0 = ker119891(34)
which end the proof
Locally we obtain
119867 = (119892119886119887
minus119901119886119901119887
1198702)119889119909119886 otimes 119889119909119887 minus (119892119886119887 minus
119901119886119901119887
1198702)120575119901119886otimes 120575119901119887
(35)
with det(119892119886119887
minus (1199011198861199011198871198702)) = 0 since (119892
119886119887minus (1199011198861199011198871198702))119901119887 =
119901119886minus 119901119886= 0 and similarly det(119892119886119887 minus (1199011198861199011198871198702)) = 0 since
(119892119886119887 minus (1199011198861199011198871198702))119901119887= 119901119886 minus 119901119886 = 0
Remark 11 The map 119867 is a transversally singular pseudo-Riemannian metric on (119876lowastF F)
5 An Almost (Para)Contact Structure onTransverse Liouville Distribution of(119876lowastFF)
Denote by 1205852 the line vector bundle over119876lowastF spanned by 120585
2
and we define the transverse vertical Liouville distribution asthe complementary orthogonal distribution 119878(119876lowastF) to 120585
2
in119881(119876lowastF)with respect to119866V namely119881(119876lowastF) = 119878(119876lowastF)oplus
1205852 Hence 119878(119876lowastF) is defined by 120572 = 1205962|
119881(119876lowastF) that is
Γ (119878 (119876lowastF)) = 119883 isin Γ (119881 (119876lowastF)) 120572 (119883) = 0 (36)
Thus any transverse vertical vector field119883 isin Γ(119881(119876lowastF)) canbe expressed as
119883 = 119875119883 + 120572 (119883) 1205852 (37)
where 119875 is the projection morphism of 119881(119876lowastF) on 119878(119876lowastF)By direct calculations one gets the following
Proposition 12 For any transverse vertical vector fields119883119884 isinΓ(119881(119876lowastF)) one has
119866V(119883 119875119884) = 119866V
(119875119883 119875119884) = 119866V(119883 119884) minus 120572 (119883) 120572 (119884) (38)
Theorem 13 The transverse vertical Liouville distribution119878(119876lowastF) is integrable
Proof The proof follows using an argument similar to Theo-rem 31 [14] (see alsoTheorem 21 [6] orTheorem 4 [15])
In the following we will consider the transverse Liouvilledistribution of (119876lowastF F) as the complementary orthogonaldistribution 119878(119876lowastF) to 120585
2 in 119876F with respect to 119866 that is
119878(119876lowastF) = 119867(119876lowastF) oplus 119878(119876lowastF)Let us restrict to 119878(119876lowastF) all the geometrical structures
introduced in Section 4 for all 119876F We indicate this byoverlines Hence we have
(i) 1205851= 1205851since 120585
1lies in 119878(119876lowastF)
(ii) 1205962 = 0 since 1205962(119883) = 119866(119883 1205852) = 0 for every
transverse vector field119883 isin 119878(119876lowastF)(iii) 119866 = 119866|
119878(119876lowastF)
(iv) 119891(119883) = 120601(119883) + 1205761205961(119883) otimes 1205852is an endomorphism of
119878(119876lowastF) since
119866(119891 (119883) 1205852) = 119866 (120601 (119883) 1205852) + 1205761205961 (119883)119866 (1205852 1205852)
= 1205962 (120601 (119883)) + 1205761205961 (119883) = 0(39)
We denote now 120585 = 1205851and 120578 = 1205961
By Theorem 8 we obtain the following
Theorem 14 The triple (119891 120585 120578) provides an almost(para)contact structure on 119878(119876lowastF) that is
(i) 1198913
+ 120576119891 = 0 119903119886119899119896119891 = 2119899 minus 2 = (2119899 minus 1) minus 1
(ii) 120578(120585) = 1 119891(120585) = 0 120578 ∘ 119891 = 0
(iii) 1198912
(119883) = minus120576(119883 minus 120578(119883)120585) for119883 isin 119878(119876lowastF)
Also by Theorem 9 we obtain the following
Theorem 15 The transversal Riemannian metric 119866 verifies
119866(119891 (119883) 119891 (119884)) = 119866 (119883 119884) minus 120578 (119883) 120578 (119884) (40)
for every transverse vector fields119883119884 isin 119878(119876lowastF)
Concluding the ensemble (119891 120585 120578 119866) is an almost(para)contact Riemannian structure on 119878(119876lowastF)
Conflict of Interests
The authors declare that they have no conflict of interests
Authorsrsquo Contribution
Both authors contributed equally to the paper All the authorsread and approved the final paper
Acknowledgment
The first author is supported by the Grant ldquoFellowship ofTransilvania University of Brasov 2014rdquo
References
[1] A Miernowski andWMozgawa ldquoLift of the Finsler foliation toits normal bundlerdquo Differential Geometry and Its Applicationsvol 24 no 2 pp 209ndash214 2006
[2] P Popescu and C Ida ldquoFoliations compatible with hamilto-niansrdquo in Proceedings of the BSG International Conference onDifferential Geometry and Dynamical Systems (DGDS rsquo13) pp147ndash155 Bucharest Romania October 2013
6 The Scientific World Journal
[3] P Popescu and M Popescu ldquoLagrangians adapted to submer-sions and foliationsrdquoDifferential Geometry and Its Applicationsvol 27 no 2 pp 171ndash178 2009
[4] P Popescu and M Popescu ldquoFoliated vector bundles andRiemannian foliationsrdquo Comptes Rendus Mathematique vol349 no 7-8 pp 445ndash449 2011
[5] P Popescu and M Popescu ldquoLagrangians and Hamiltonianstructures and foliationsrdquo Physics AUC vol 21 no 1 pp 203ndash206 2011
[6] C Ida ldquoSome framed 119891-structures on transversally Finsler foli-ationsrdquo Annales Universitatis Mariae Curie-Sklodowska SectioA Mathematica vol 65 no 1 pp 87ndash96 2011
[7] M Abate and G Patrizio Finsler MetricsmdashA Global Approachvol 1591 of Lecture Notes in Mathematics Springer BerlinGermany 1994
[8] R Miron D Hrimiuc H Shimada and S V SabauThe Geom-etry of Hamilton and Lagrange Spaces vol 118 of FundamentalTheories of Physics Kluwer Academic 2001
[9] K Yano ldquoOn a structure defined by a tensor field of type (1 1)satisfying 1198913 + 119891 = 0rdquo Tensor vol 14 pp 99ndash109 1963
[10] M Anastasiei ldquoA framed f-structure on tangent bundle of aFinsler spacerdquoAnalele Universitatii Bucuresti SeriaMatematicavol 49 pp 3ndash9 2000
[11] M Gırtu ldquoAn almost 2-paracontact structure on the cotangentbundle of a Cartan spacerdquoHacettepe Journal ofMathematics andStatistics vol 33 pp 15ndash22 2004
[12] M Gırtu ldquoA Sasakian structure on the indicatrix bundle of aCartan spacerdquo Analele Stiintifice ale Universitatii ldquoAlICuzardquo dinIasi Tomul LIII vol 1 pp 177ndash186 2007
[13] I Mihai R Rosca and L Verstraelen Some Aspects of theDifferential Geometry of Vector Fields vol 2 of Centre forPure and Applied Differential Geometry (PADGE) KatholiekeUniversiteit Leuven Leuven Belgium 1996
[14] A Bejancu and H R Farran ldquoOn the vertical bundle of apseudo-Finsler manifoldrdquo International Journal of Mathematicsand Mathematical Sciences vol 22 no 3 pp 637ndash642 1999
[15] C Ida and A Manea ldquoA vertical Liouville subfoliation on thecotangent bundle of a Cartan space and some related struc-turesrdquo International Journal of Geometric Methods in ModernPhysics vol 11 no 6 21 pages 2014
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
The Scientific World Journal 5
Proof The symmetry and bilinearity are obvious Also thenull space of119867 is
119883 isin Γ (119876F) | 119867 (119883 119884) = 0 forall119884 isin Γ (119876F)
= 119883 isin Γ (119876F) | 119866 (119891 (119883) 119884) = 0 = ker119891(34)
which end the proof
Locally we obtain
119867 = (119892119886119887
minus119901119886119901119887
1198702)119889119909119886 otimes 119889119909119887 minus (119892119886119887 minus
119901119886119901119887
1198702)120575119901119886otimes 120575119901119887
(35)
with det(119892119886119887
minus (1199011198861199011198871198702)) = 0 since (119892
119886119887minus (1199011198861199011198871198702))119901119887 =
119901119886minus 119901119886= 0 and similarly det(119892119886119887 minus (1199011198861199011198871198702)) = 0 since
(119892119886119887 minus (1199011198861199011198871198702))119901119887= 119901119886 minus 119901119886 = 0
Remark 11 The map 119867 is a transversally singular pseudo-Riemannian metric on (119876lowastF F)
5 An Almost (Para)Contact Structure onTransverse Liouville Distribution of(119876lowastFF)
Denote by 1205852 the line vector bundle over119876lowastF spanned by 120585
2
and we define the transverse vertical Liouville distribution asthe complementary orthogonal distribution 119878(119876lowastF) to 120585
2
in119881(119876lowastF)with respect to119866V namely119881(119876lowastF) = 119878(119876lowastF)oplus
1205852 Hence 119878(119876lowastF) is defined by 120572 = 1205962|
119881(119876lowastF) that is
Γ (119878 (119876lowastF)) = 119883 isin Γ (119881 (119876lowastF)) 120572 (119883) = 0 (36)
Thus any transverse vertical vector field119883 isin Γ(119881(119876lowastF)) canbe expressed as
119883 = 119875119883 + 120572 (119883) 1205852 (37)
where 119875 is the projection morphism of 119881(119876lowastF) on 119878(119876lowastF)By direct calculations one gets the following
Proposition 12 For any transverse vertical vector fields119883119884 isinΓ(119881(119876lowastF)) one has
119866V(119883 119875119884) = 119866V
(119875119883 119875119884) = 119866V(119883 119884) minus 120572 (119883) 120572 (119884) (38)
Theorem 13 The transverse vertical Liouville distribution119878(119876lowastF) is integrable
Proof The proof follows using an argument similar to Theo-rem 31 [14] (see alsoTheorem 21 [6] orTheorem 4 [15])
In the following we will consider the transverse Liouvilledistribution of (119876lowastF F) as the complementary orthogonaldistribution 119878(119876lowastF) to 120585
2 in 119876F with respect to 119866 that is
119878(119876lowastF) = 119867(119876lowastF) oplus 119878(119876lowastF)Let us restrict to 119878(119876lowastF) all the geometrical structures
introduced in Section 4 for all 119876F We indicate this byoverlines Hence we have
(i) 1205851= 1205851since 120585
1lies in 119878(119876lowastF)
(ii) 1205962 = 0 since 1205962(119883) = 119866(119883 1205852) = 0 for every
transverse vector field119883 isin 119878(119876lowastF)(iii) 119866 = 119866|
119878(119876lowastF)
(iv) 119891(119883) = 120601(119883) + 1205761205961(119883) otimes 1205852is an endomorphism of
119878(119876lowastF) since
119866(119891 (119883) 1205852) = 119866 (120601 (119883) 1205852) + 1205761205961 (119883)119866 (1205852 1205852)
= 1205962 (120601 (119883)) + 1205761205961 (119883) = 0(39)
We denote now 120585 = 1205851and 120578 = 1205961
By Theorem 8 we obtain the following
Theorem 14 The triple (119891 120585 120578) provides an almost(para)contact structure on 119878(119876lowastF) that is
(i) 1198913
+ 120576119891 = 0 119903119886119899119896119891 = 2119899 minus 2 = (2119899 minus 1) minus 1
(ii) 120578(120585) = 1 119891(120585) = 0 120578 ∘ 119891 = 0
(iii) 1198912
(119883) = minus120576(119883 minus 120578(119883)120585) for119883 isin 119878(119876lowastF)
Also by Theorem 9 we obtain the following
Theorem 15 The transversal Riemannian metric 119866 verifies
119866(119891 (119883) 119891 (119884)) = 119866 (119883 119884) minus 120578 (119883) 120578 (119884) (40)
for every transverse vector fields119883119884 isin 119878(119876lowastF)
Concluding the ensemble (119891 120585 120578 119866) is an almost(para)contact Riemannian structure on 119878(119876lowastF)
Conflict of Interests
The authors declare that they have no conflict of interests
Authorsrsquo Contribution
Both authors contributed equally to the paper All the authorsread and approved the final paper
Acknowledgment
The first author is supported by the Grant ldquoFellowship ofTransilvania University of Brasov 2014rdquo
References
[1] A Miernowski andWMozgawa ldquoLift of the Finsler foliation toits normal bundlerdquo Differential Geometry and Its Applicationsvol 24 no 2 pp 209ndash214 2006
[2] P Popescu and C Ida ldquoFoliations compatible with hamilto-niansrdquo in Proceedings of the BSG International Conference onDifferential Geometry and Dynamical Systems (DGDS rsquo13) pp147ndash155 Bucharest Romania October 2013
6 The Scientific World Journal
[3] P Popescu and M Popescu ldquoLagrangians adapted to submer-sions and foliationsrdquoDifferential Geometry and Its Applicationsvol 27 no 2 pp 171ndash178 2009
[4] P Popescu and M Popescu ldquoFoliated vector bundles andRiemannian foliationsrdquo Comptes Rendus Mathematique vol349 no 7-8 pp 445ndash449 2011
[5] P Popescu and M Popescu ldquoLagrangians and Hamiltonianstructures and foliationsrdquo Physics AUC vol 21 no 1 pp 203ndash206 2011
[6] C Ida ldquoSome framed 119891-structures on transversally Finsler foli-ationsrdquo Annales Universitatis Mariae Curie-Sklodowska SectioA Mathematica vol 65 no 1 pp 87ndash96 2011
[7] M Abate and G Patrizio Finsler MetricsmdashA Global Approachvol 1591 of Lecture Notes in Mathematics Springer BerlinGermany 1994
[8] R Miron D Hrimiuc H Shimada and S V SabauThe Geom-etry of Hamilton and Lagrange Spaces vol 118 of FundamentalTheories of Physics Kluwer Academic 2001
[9] K Yano ldquoOn a structure defined by a tensor field of type (1 1)satisfying 1198913 + 119891 = 0rdquo Tensor vol 14 pp 99ndash109 1963
[10] M Anastasiei ldquoA framed f-structure on tangent bundle of aFinsler spacerdquoAnalele Universitatii Bucuresti SeriaMatematicavol 49 pp 3ndash9 2000
[11] M Gırtu ldquoAn almost 2-paracontact structure on the cotangentbundle of a Cartan spacerdquoHacettepe Journal ofMathematics andStatistics vol 33 pp 15ndash22 2004
[12] M Gırtu ldquoA Sasakian structure on the indicatrix bundle of aCartan spacerdquo Analele Stiintifice ale Universitatii ldquoAlICuzardquo dinIasi Tomul LIII vol 1 pp 177ndash186 2007
[13] I Mihai R Rosca and L Verstraelen Some Aspects of theDifferential Geometry of Vector Fields vol 2 of Centre forPure and Applied Differential Geometry (PADGE) KatholiekeUniversiteit Leuven Leuven Belgium 1996
[14] A Bejancu and H R Farran ldquoOn the vertical bundle of apseudo-Finsler manifoldrdquo International Journal of Mathematicsand Mathematical Sciences vol 22 no 3 pp 637ndash642 1999
[15] C Ida and A Manea ldquoA vertical Liouville subfoliation on thecotangent bundle of a Cartan space and some related struc-turesrdquo International Journal of Geometric Methods in ModernPhysics vol 11 no 6 21 pages 2014
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
6 The Scientific World Journal
[3] P Popescu and M Popescu ldquoLagrangians adapted to submer-sions and foliationsrdquoDifferential Geometry and Its Applicationsvol 27 no 2 pp 171ndash178 2009
[4] P Popescu and M Popescu ldquoFoliated vector bundles andRiemannian foliationsrdquo Comptes Rendus Mathematique vol349 no 7-8 pp 445ndash449 2011
[5] P Popescu and M Popescu ldquoLagrangians and Hamiltonianstructures and foliationsrdquo Physics AUC vol 21 no 1 pp 203ndash206 2011
[6] C Ida ldquoSome framed 119891-structures on transversally Finsler foli-ationsrdquo Annales Universitatis Mariae Curie-Sklodowska SectioA Mathematica vol 65 no 1 pp 87ndash96 2011
[7] M Abate and G Patrizio Finsler MetricsmdashA Global Approachvol 1591 of Lecture Notes in Mathematics Springer BerlinGermany 1994
[8] R Miron D Hrimiuc H Shimada and S V SabauThe Geom-etry of Hamilton and Lagrange Spaces vol 118 of FundamentalTheories of Physics Kluwer Academic 2001
[9] K Yano ldquoOn a structure defined by a tensor field of type (1 1)satisfying 1198913 + 119891 = 0rdquo Tensor vol 14 pp 99ndash109 1963
[10] M Anastasiei ldquoA framed f-structure on tangent bundle of aFinsler spacerdquoAnalele Universitatii Bucuresti SeriaMatematicavol 49 pp 3ndash9 2000
[11] M Gırtu ldquoAn almost 2-paracontact structure on the cotangentbundle of a Cartan spacerdquoHacettepe Journal ofMathematics andStatistics vol 33 pp 15ndash22 2004
[12] M Gırtu ldquoA Sasakian structure on the indicatrix bundle of aCartan spacerdquo Analele Stiintifice ale Universitatii ldquoAlICuzardquo dinIasi Tomul LIII vol 1 pp 177ndash186 2007
[13] I Mihai R Rosca and L Verstraelen Some Aspects of theDifferential Geometry of Vector Fields vol 2 of Centre forPure and Applied Differential Geometry (PADGE) KatholiekeUniversiteit Leuven Leuven Belgium 1996
[14] A Bejancu and H R Farran ldquoOn the vertical bundle of apseudo-Finsler manifoldrdquo International Journal of Mathematicsand Mathematical Sciences vol 22 no 3 pp 637ndash642 1999
[15] C Ida and A Manea ldquoA vertical Liouville subfoliation on thecotangent bundle of a Cartan space and some related struc-turesrdquo International Journal of Geometric Methods in ModernPhysics vol 11 no 6 21 pages 2014
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
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
![684.8 [1,1]](https://static.fdocuments.in/doc/165x107/5681611b550346895dd07429/6848-11.jpg)
