Hindawi Publishing CorporationJournal of MathematicsVolume 2013 Article ID 395787 7 pageshttpdxdoiorg1011552013395787

Research ArticleTotally Contact Umbilical Lightlike Hypersurfaces ofIndenite119982119982-Manifolds

Rachna Rani1 Rakesh Kumar2 and R K Nagaich3

1 University College Moonak Punjab 148033 India2University College of Engineering Punjabi University Patiala 147002 India3Department of Mathematics Punjabi University Patiala 147002 India

Correspondence should be addressed to Rakesh Kumar dr_rk37cyahoocoin

Received 22 August 2012 Revised 5 November 2012 Accepted 8 November 2012

Academic Editor Mingsheng Liu

Copyright copy 2013 Rachna Rani et al is 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

e study totally contact umbilical lightlike hypersurfaces of indenite 119982119982-manifolds and prove the nonexistence of totally contactumbilical lightlike hypersurface in indenite 119982119982-space form

1 Introduction

e general theory of nondegenerate submanifolds of Rie-mannian or semi-Riemannian manifolds is one of the mostimportant topics of differential geometry [1 2] But thetheory of degenerate or lightlike submanifolds of semi-Riemannian manifolds is relatively new [3] and different dueto the fact that their normal vector bundle intersects withthe tangent bundle us the study of lightlike submanifoldsbecomes more difficult than the study of nondegeneratesubmanifolds and one cannot use in the usual way theclassical theory of submanifold to dene induced objects on alightlike submanifold e degenerate geometry rises withinthe semi-Riemannian context due to the existence of causalcharacter of geometrical objects their spacelike timelikeor lightlike nature implies the existence of three types ofhypersurfaces and submanifolds

A lightlike framed hypersurface of a Lorentz119862119862-manifoldwith an inducedmetric connection is a Killing horizon and aglobally hyperbolic spacetime a de Sitter spacetime can carrya framed structure [3 4] Moreover the contact geometry hasa signicant use in differential euations optics and phasespaces of dynamical systems [5ndash7] All these motivated us tostudy lightlike hypersurfaces of indenite globally framed 119891119891-manifold in particular indenite 119982119982-manifolds

In this paper we study totally contact umbilical light-like hypersurfaces of indenite 119982119982-manifolds and prove the

nonexistence of totally contact umbilical lightlike hypersur-face in indenite 119982119982-space form

2 Preliminaries

Amanifold119872119872 of dimension 2119899119899119899119899119899 is called a globally framed119891119891-manifold (119892119892119892119891119891119892119891119891-manifold) if it is endowedwith a nowherevanishing (1 1)-tensor eld120601120601 of constant rank such that ker120601120601is parallelizable that is there exist global vector elds 120577120577120572120572 120572120572 1205721hellip 119899119899119903 with their dual 1-forms 120578120578120572120572 satisfying

1206011206012= minus119868119868 119899 120578120578120572120572 otimes 120577120577120572120572 120578120578120572120572 1007650100765012057712057712057312057310076661007666 = 120575120575

120572120572120573120573119892 (1)

If the metric 119892119892 is a semi-Riemannian metric with index 1205841205840 lt 120584120584 lt 2119899119899 119899 119899119899 such that

119892119892 10076501007650120601120601120601120601 12060112060112060112060110076661007666 = 119892119892 (120601120601 120601120601) minus11989911989910055761005576120572120572=1

120598120598120572120572120578120578120572120572 (120601120601) 120578120578120572120572 (120601120601) (2)

for any120601120601120601120601 120572 119883(119883119883119872119872) 120598120598120572120572 = plusmn1 accordingly 120577120577120572120572 is either space-like or timelike then the 119892119892119892119891119891119892119891119891-manifold (1198721198722119899119899119899119899119899 120601120601 120577120577120572120572 120578120578

120572120572)is called an indenite metric 119892119892119892119891119891119892119891119891-manifold en clearly120578120578120572120572(120601120601) = 120598120598120572120572119892119892(120601120601 120577120577120572120572) for any 120572120572 120572 1hellip 119899119899119903 An indenitemetric 119892119892119892119891119891119892119891119891-manifold is called an indenite 119982119982-manifold ifit is normal that is the tensor eld 119873119873 = 119873119873120601120601 119899 2119889119889120578120578120572120572 otimes 120577120577120572120572

2 Journal of Mathematics

vanishes where 119873119873120601120601 is Nijenhuis tensor of 120601120601 and 119889119889120578120578120572120572 = Φfor any 120572120572 120572 120572120572120572120572 120572 120572120572120572 where Φ(119883119883120572119883119883119883 = 119892119892(119883119883120572 120601120601119883119883119883 for any119883119883120572119883119883 120572 119883(119883119883119872119872119883 e Levi-Civita connection of an indenite119982119982-manifold satises

10076501007650nabla11988311988312060112060110076661007666119883119883 = 119892119892 10076501007650120601120601119883119883120572 12060112060111988311988310076661007666 120577120577 120577 120578120578 (119883119883119883 1206011206012(119883119883119883 120572 (3)

where 120577120577 = 120577120572120572120572120572=120572 120577120577120572120572 and 120578120578 = 120577

120572120572120572120572=120572 120598120598120572120572120578120578

120572120572 Also

nabla119883119883120577120577120572120572 = minus120598120598120572120572120601120601119883119883120572 (4)

and ker120601120601 is integrable at distribution since nabla120577120577120572120572120577120577120573120573 = 0 fordetail see [8]

We recall notations and fundamental equations for light-like hypersurfaces which are due to Duggal and Bejancu [3]

Let (119872119872120572 119892119892119883 be a (2119899119899 120577 120572119883-dimensional semi-Riemannianmanifold with index 119904119904120572 0 119904 119904119904 119904 2119899119899 120577 120572 and let (119872119872120572 119892119892119883be a hypersurface of 119872119872 with 119892119892 = 119892119892119892119872119872 119872119872 is a lightlikehypersurface of 119872119872 if 119892119892 is of constant rank 2119899119899 minus 120572 and thenormal bundle 119883119883119872119872⟂ is a distribution of rank 120572 on 119872119872 Acomplementary bundle of 119883119883119872119872⟂ in 119883119883119872119872 is a rank 2119899119899 minus 120572nondegenerate distribution over119872119872 It is denoted by 119878119878(119883119883119872119872119883and known as a screen distribution erefore we have

119883119883119872119872 = 119878119878 (119883119883119872119872119883 ⟂ 119883119883119872119872⟂120572

11988311988311987211987210092501009250119872119872 = 119878119878 (119883119883119872119872119883 ⟂ 119878119878(119883119883119872119872119883⟂120572(5)

where 119878119878(119883119883119872119872119883⟂ is a orthogonal complementary vector bundleof 119878119878(119883119883119872119872119883 in 119883119883119872119872119892119872119872 e following theorem has importantroles in studying the geometry of lightlike hypersurface

eorem1 Let (119872119872120572 119892119892120572 119878119878(119883119883119872119872119883119883 be a lightlike hypersurface of asemi-Riemannian manifold (119872119872120572 119892119892119883 en there exists a uniquevector bundle tr(119883119883119872119872119883 of rank 120572 over 119872119872 such that for anynonzero section 120585120585 of 119883119883119872119872⟂ on a coordinate neighborhood119984119984 119984119872119872 there exists a unique section119873119873 of tr(119883119883119872119872119883 on119984119984 satisfying

119892119892 (120585120585120572119873119873119883 = 120572120572 119892119892 (119873119873120572119873119873119883 = 119892119892 (119873119873120572119873119873119883 = 0120572

forall119873119873 120572 119883 10076491007649119878119878 (11988311988311987211987211988311989211998411998410076651007665 (6)

Hence we have the following decompositions of 119883119883119872119872119892119872119872

11988311988311987211987210092501009250119872119872 = 119878119878 (119883119883119872119872119883 ⟂ 10076501007650119883119883119872119872⟂ oplus tr (11988311988311987211987211988310076661007666

= 119883119883119872119872 oplus tr (119883119883119872119872119883 (7)

Let nabla be the Levi-Civita connection on 119872119872 with respectto 119892119892 en using the decompositions in (7) Gauss andWeingarten formulae are given as

nabla119883119883119883119883 = nabla119883119883119883119883 120577 119884 (119883119883120572 119883119883119883 120572

nabla119883119883119873119873 = minus119873119873119873119873119883119883 120577 nabla119905119905119883119883119873119873120572(8)

for any 119883119883120572119883119883 120572 119883(119883119883119872119872119883 and 119873119873 120572 119883(tr(119883119883119872119872119883119883 where nabla119883119883119883119883and 119873119873119873119873119883119883 belong to 119883(119883119883119872119872119883 while 119884(119883119883120572 119883119883119883 and nabla119905119905119883119883119873119873 belongto 119883(tr(119883119883119872119872119883119883 Here nabla is a torsion free linear connection

on 119872119872 119884 is a 119883(tr(119883119883119872119872119883119883-valued symmetric bilinear form on119883(119883119883119872119872119883 and known as the second fundamental form 119873119873119873119873 is alinear operator on 119883(119883119883119872119872119883 and known as the shape operatorof lightlike immersion and nabla119905119905 is a linear connection on119883(tr(119883119883119872119872119883119883

Locally for the pair 120572120585120585120572119873119873120572 following the Duggal andBejancursquos notation [3] we recall local second fundamentalform 119861119861 and 120572-form 120591120591 as

119861119861 (119883119883120572119883119883119883 = 119892119892 (119884 (119883119883120572 119883119883119883 120572 120585120585119883 120572

therefore 119884 (119883119883120572 119883119883119883 = 119861119861 (119883119883120572 119883119883119883119873119873

120591120591 (119883119883119883 = 119892119892 10076501007650nabla119905119905119883119883119873119873120572 12058512058510076661007666

therefore nabla119905119905119883119883119873119873 = 120591120591 (119883119883119883119873119873

(9)

Hence locally (8) becomes

nabla119883119883119883119883 = nabla119883119883119883119883 120577 119861119861 (119883119883120572119883119883119883119873119873120572

nabla119883119883119873119873 = minus119873119873119873119873119883119883 120577 120591120591 (119883119883119883119873119873(10)

If 119875119875 denotes the projection morphism of 119883119883119872119872 on 119878119878(119883119883119872119872119883then from (5) we have

nabla119883119883119875119875119883119883 = nablalowast119883119883119875119875119883119883 120577 119884lowast (119883119883120572 119875119875119883119883119883 120572

nabla119883119883120585120585 = minus119873119873lowast120585120585119883119883 120577 nablalowast119905119905119883119883 120585120585120572

(11)

for any 119883119883120572119883119883 120572 119883(119883119883119872119872119883 120585120585 120572 119883(119883119883119872119872⟂119883 where nablalowast119883119883119875119875119883119883and 119873119873lowast

120585120585119883119883 belong to 119883(119878119878(119883119883119872119872119883119883 while 119884lowast(119883119883120572 119875119875119883119883119883 and nablalowast119905119905119883119883 120585120585belong to 119883(119883119883119872119872⟂119883 Here 119884lowast and 119873119873lowast

120585120585 are called the secondfundamental form and the shape operator of the screendistribution respectively It should be noted that the inducedlinear connection nabla is not a metric connection as it satises

10076491007649nabla11988311988311989211989210076651007665 (119883119883120572119884119884119883 = 119861119861 (119883119883120572 119883119883119883 120579120579 (119884119884119883 120577 119861119861 (119883119883120572119884119884119883 120579120579 (119883119883119883 120572 (12)

where 120579120579 is a differential 120572 form locally dened on119872119872 by 120579120579(120579119883 =119892119892(119873119873120572 120579119883 By direct calculation we have

119892119892 10076491007649119873119873119873119873119883119883120572 11987511987511988311988310076651007665 = 119892119892 10076491007649119873119873120572 119884lowast (119883119883120572 11987511987511988311988311988310076651007665 120572 119892119892 1007649100764911987311987311987311987311988311988312057211987311987310076651007665 = 0120572

119892119892 10076501007650119873119873lowast120585120585119883119883120572 11987511987511988311988310076661007666 = 119892119892 (120585120585120572 119884 (119883119883120572 119875119875119883119883119883119883 120572 119892119892 10076501007650119873119873lowast

12058512058511988311988312057211987311987310076661007666 = 0120572(13)

for any 119883119883120572119883119883 120572 119883(119883119883119872119872119883 120585120585 120572 119883(119883119883119872119872⟂119883 and 119873119873 120572 119883(tr(119883119883119872119872119883119883According to Duggal and Bejancursquos notation [3] locally wehave know

119862119862 (119883119883120572 119875119875119883119883119883 = 119892119892 10076491007649119884lowast (119883119883120572 119875119875119883119883119883 12057211987311987310076651007665

therefore 119884lowast (119883119883120572 119875119875119883119883119883 = 119862119862 (119883119883120572 119875119875119883119883119883 120585120585120572

120598120598 (119883119883119883 = 119892119892 10076501007650nablalowast119905119905119883119883 12058512058512057211987311987310076661007666

therefore nablalowast119905119905119883119883 120585120585 = 120598120598 (119883119883119883 120585120585

(14)

en (11) become

nabla119883119883119875119875119883119883 = nablalowast119883119883119875119875119883119883 120577 119862119862 (119883119883120572 119875119875119883119883119883 120585120585120572 (15)

nabla119883119883120585120585 = minus119873119873lowast120585120585119883119883 120577 120598120598 (119883119883119883 120585120585 = minus119873119873lowast

120585120585119883119883 minus 120591120591 (119883119883119883 120585120585 (16)

Journal of Mathematics 3

and also give

119892119892 10076491007649119860119860119873119873119884119884119884 11988411988411988411988410076651007665 = 119862119862 (119884119884119884 119884119884119884119884) 119884 119892119892 1007649100764911986011986011987311987311988411988411988411987311987310076651007665 = 0119884

119892119892 10076501007650119860119860lowast120585120585119883119883119884 11988411988411988411988410076661007666 = 119861119861 (119883119883119884 119884119884119884119884) 119884 119892119892 10076501007650119860119860lowast

12058512058511988311988311988411987311987310076661007666 = 0(17)

From (9) it is clear that

119861119861 (119883119883119884 120585120585) = 0119884 (18)

that is the second fundamental form of a lightlikehypersurface is degenerate Following the notations ofDuggal and Bejancu [3] for the curvature tensor 119877119877 of119872119872 we have 119877119877(119883119883119884119884119884119884119877119877119884119884119884) = 119892119892(119877119877(119883119883119884 119884119884)119877119877119884119884119884) for any119883119883119884119884119884119884119877119877119884119884119884 119883 119883(119883119883119872119872)

3 Lightlike Hypersurface ofndenite119982119982-Manifolds

Let (119872119872119884 119892119892119884 119872119872(119883119883119872119872)) be a lightlike hypersurface of an indenite119892119892119892119892119892119892-manifold (1198721198722119899119899119899119899119899119884 120601120601119884 120577120577120572120572119884 120578120578

120572120572119884 119892119892) and 120572120572 119883 120572120572119884120572 119884 119899119899120572 suchthat the characteristic vector elds 120577120577120572120572 120572120572 119883 120572120572119884120572 119884 119899119899120572 aretangent to119872119872 As the ambient manifold119872119872 has an additionalgeometric structure 120601120601 we must look for a particular screendistribution on 119872119872 Since 119892119892(120601120601120585120585119884 120585120585) = 0 for any 120585120585 119883 119883(119883119883119872119872⟂)therefore 120601120601(119883119883119872119872⟂) is a distribution on119872119872 of rank 120572 such that119883119883119872119872⟂ cap120601120601(119883119883119872119872⟂) = 1205720120572 Moreover 119892119892(120601120601119873119873119884 120585120585) = 120601119892119892(119873119873119884 120601120601120585120585) = 0for any119873119873 119883 119883(119873119873(119883119883119872119872)) therefore 120601120601119873119873 is tangent to119872119872 Since119892119892(120601120601119873119873119884119873119873) = 0 that is the component of 120601120601119873119873 with respectto 120585120585 vanishes this implies 120601120601119873119873 119883 119883(119872119872(119883119883119872119872)) As 120585120585 and 119873119873 arenull vector elds satisfying 119892119892(120585120585119884119873119873) = 120572 therefore from (2)we deduce that 120601120601120585120585 and 120601120601119873119873 are null vector elds satisfying119892119892(120601120601120585120585119884 120601120601119873119873) = 120572 Hence 120601120601(119883119883119872119872⟂) oplus 120601120601(119873119873(119883119883119872119872)) is a vectorsubbundle of 119872119872(119883119883119872119872) of rank 2 It is known [9] that if structurevector elds 120577120577120572120572 120572120572 119883 120572120572120572 119884 119899119899120572 are tangent to119872119872 then 120577120577120572120572 belongto 119872119872(119883119883119872119872) erefore 119892119892(120601120601120585120585119884 120577120577120572120572) = 119892119892(120601120601119873119873119884 120577120577120572120572) = 0 implies thatthere exists a nondegenerate distribution 1198631198630 of rank 2119899119899 120601 119899on119872119872 such that

119872119872 (119883119883119872119872) = 10076821007682120601120601 (119873119873 (119883119883119872119872)) oplus 120601120601119883119883119872119872⟂10076981007698 ⟂ 1198631198630 ⟂ 1007682100768212057712057712057212057210076981007698 119884 (19)

therefore

119883119883119872119872 = 10076821007682120601120601 (119873119873 (119883119883119872119872)) oplus 120601120601119883119883119872119872⟂10076981007698 ⟂ 1198631198630 ⟂ 1007682100768212057712057712057212057210076981007698 ⟂ 119883119883119872119872⟂(20)

Let 119863119863 = 119883119883119872119872⟂ ⟂ 120601120601119883119883119872119872⟂ ⟂ 1198631198630 and let 119863119863119863 = 120601120601 119873119873(119883119883119872119872) then119863119863 clearly is invariant and119863119863119863 is anti-invariant under 120601120601 and wehave

119883119883119872119872 = 119863119863119863 oplus 119863119863 ⟂ 1007682100768212057712057712057212057210076981007698 (21)

onsider the local lightlike vector elds as

119880119880 = 120601120601120601119873119873 119883 120601120601 (119873119873 (119883119883119872119872)) = 119863119863119863119884

119881119881 = 120601120601120601120585120585 119883 119863119863(22)

Using the decompositions in (21) any 119883119883 119883 119883(119883119883119872119872) can bewritten as

119883119883 = 119872119872119883119883 119899 119883119883119883119883 11989911989911989910055761005576120572120572=120572

120578120578120572120572 (119883119883) 120577120577120572120572119884 (23)

where 119872119872 and 119883119883 are the projection morphisms of 119883119883119872119872 into 119863119863and119863119863119863 respectively andwhere119883119883119883119883 = 119876119876(119883119883)119880119880 and 119876119876 is a local120572-form on119872119872 dened by 119876119876(119883119883) = 119892119892(119881119881119884119883119883) erefore

119876119876 (119880119880) = 120572119884 119876119876 (119883119883) = 0119884 forall119883119883 119883 119883 (119863119863) 119884 1206011206012119873119873 = 120601119873119873 (24)

Applying 120601120601 to (23) and then using (24) we obtain 120601120601119883119883 =120601120601(119872119872119883119883) 119899 119876119876(119883119883)119873119873 By assuming 120601120601(119872119872119883119883) = 120601120601119883119883 for any 119883119883 119883119883(119883119883119872119872) we obtain a tensor eld 120601120601 of type (120572119884 120572) on 119872119872 andgiven by

120601120601119883119883 = 120601120601119883119883 119899 119876119876 (119883119883)119873119873 (25)

Applying 120601120601 to (25) and then using the denition of 120601120601119883119883 weobtain

1206011206012119883119883 = 120601119883119883 119899 119876119876 (119883119883)119880119880 11989911989911989910055761005576120572120572=120572

120578120578120572120572 (119883119883) 120577120577120572120572 (26)

Moreover since 120601120601119883119883 = 120601120601(119872119872119883119883) therefore

120601120601119880119880 = 0119884 120578120578120572120572 ∘ 120601120601 = 0119884 119876119876 1007649100764912060112060111988311988310076651007665 = 0119884 (27)

for any 119883119883 119883 119883(119883119883119872119872) us we have the following theoremanalogous to a theorem proved in [10]

eorem 2 Let (119872119872119884 120601120601119884 120577120577120572120572119884 120578120578120572120572119884 119892119892) be an indenite 119892119892119892119892119892119892-

manifold and let (119872119872119884 119892119892119884 119872119872(119883119883119872119872)) be a lightlike hypersurface of119872119872 then (119872119872119884 120601120601119884 120577120577120572120572119884 119880119880119884 120578120578

120572120572119884 119876119876) is also a 119892119892119892119892119892119892-manifold

Since 120601120601119880119880 = 0 therefore applying 120601120601 to (26) we get 1206011206013 119899120601120601 =0 this implies that 120601120601 is an 119892119892 structure of constant rank on119872119872Now using (2) and (25) we obtain

119892119892 10076491007649120601120601119883119883119884 12060112060111988411988410076651007665 = 119892119892 (119883119883119884 119884119884) 120601 119876119876 (119883119883) 120584120584 (119884119884)

120601 119876119876 (119884119884) 120584120584 (119883119883) 12060111989911989910055761005576120572120572=120572

120598120598120572120572120578120578120572120572 (119883119883) 120578120578120572120572 (119884119884) 119884

(28)

for any119883119883119884119884119884 119883 119883(119883119883119872119872) where 120584120584 is a 120572 form locally dened on119872119872 by 120584120584(120584) = 119892119892(119880119880119884 120584) Replace119883119883 by 120601120601119883119883 in (28) and using (26)and (27) we derive

119892119892 1007649100764912060112060111988311988311988411988411988410076651007665 119899 119892119892 10076491007649119883119883119884 12060112060111988411988410076651007665

= 120601119876119876 (119883119883) 120579120579 (119884119884) 120601 119876119876 (119884119884) 120579120579 (119883119883) (29)

Let 119872119872 be a lightlike hypersurface of an indenite 119982119982-manifold119872119872 then by using (4) (10) and (25) we obtain

nabla119883119883120577120577120572120572 119899 119861119861 10076501007650119883119883119884 12057712057712057212057210076661007666119873119873 = 120601120598120598120572120572120601120601119883119883 120601 120598120598120572120572119876119876 (119883119883)119873119873119884 (30)

4 Journal of Mathematics

for any 119883119883 119883 119883119883119883119883119872119872119872 then comparing the tangential andtransversal components we get

nabla119883119883120577120577120572120572 = minus120598120598120572120572120601120601119883119883120601 (31)

119861119861 10076501007650119883119883120601 12057712057712057212057210076661007666 = minus120598120598120572120572119906119906 119883119883119883119872 120601

this implies ℎ 10076501007650119883119883120601 12057712057712057212057210076661007666 = minus120598120598120572120572119906119906 119883119883119883119872119873119873119873(32)

Using (15) and (31) we get nablalowast119883119883120577120577120572120572 + 119862119862119883119883119883120601 120577120577120572120572119872120585120585 = minus120598120598120572120572120601120601119883119883taking inner product with119873119873 119883 119883119883119873119873119883119883119883119872119872119872119872 we get119862119862119883119883119883120601 120577120577120572120572119872 =minus120598120598120572120572119892119892119883120601120601119883119883120601119873119873119872 and then using (29) we obtain

11986211986210076501007650119883119883120601 12057712057712057212057210076661007666 = 120598120598120572120572119892119892 10076491007649119883119883120601 12060112060111987311987310076651007665 + 119906119906 119883119883119883119872 120579120579 119883119873119873119872 minus 119906119906 119883119873119873119872 120579120579 119883119883119883119872

= minus120598120598120572120572119892119892 119883119883119883120601119883119883119872

= minus120598120598120572120572120584120584 119883119883119883119872 119873

(33)

Using (3) for any119873119873 119883 119883119883119873119873119883119883119883119872119872119872119872 we get nabla119883119883120601120601119873119873 = 120601120601nabla119883119883119873119873 +119892119892119883119883119883120601119873119873119872120577120577 therefore

119861119861 119883119883119883120601119883119883119872 = 119892119892 119883ℎ 119883119883119883120601119883119883119872 120601 120585120585119872

= minus119892119892 10076501007650nabla119883119883120601120601119873119873120601 12058512058510076661007666

= 119892119892 10076501007650nabla119883119883120601 12060112060112058512058510076661007666

= 119892119892 1007649100764911986011986011987311987311988311988312060111988311988310076651007665

= 119862119862 119883119883119883120601119883119883119872 119873

(34)

4 Totally Contact Umbilical Lightlikeyesface of ndenite119982119982-Manifolds

ere wewill follow eancus denition [11] of totally contactumbilical submanifolds of Sasakian manifolds to state totallycontact umbilical lightlike hypersurfaces of indenite 119982119982-manifolds

e A lightlike hypersurface 119872119872 of an indenite119982119982-manifold is said to be totally contact umbilical lightlikehypersurface if the second fundamental form ℎ of119872119872 satises

ℎ 119883119883119883120601 119883119883119872 = 10076871007687119892119892 119883119883119883120601119883119883119872 minus11990311990310055761005576120572120572=120572

120598120598120572120572120578120578120572120572 119883119883119883119872 120578120578120572120572 11988311988311988311987210077031007703119867119867

+11990311990310055761005576120572120572=120572

120578120578120572120572 119883119883119883119872 ℎ 10076501007650119883119883120601 12057712057712057212057210076661007666

+11990311990310055761005576120572120572=120572

120578120578120572120572 119883119883119883119872 ℎ 10076501007650119883119883120601 12057712057712057212057210076661007666 120601

(35)

for any 119883119883120601119883119883 119883 119883119883119883119883119872119872119872 where119867119867 is a transversal vector eldon 119872119872 that is 119867119867 = 119867119867119873119873 where 119867119867 is a smooth function on119984119984 119984 119872119872

Remark 4 We can also write (35) asℎ 119883119883119883120601 119883119883119872 = 119861119861 119883119883119883120601 119883119883119872119873119873

= 10076811007681119861119861120572 119883119883119883120601 119883119883119872 + 1198611198612 119883119883119883120601 11988311988311987210076971007697119873119873120601(36)

where

119861119861120572 119883119883119883120601 119883119883119872 = 11986711986710076871007687119892119892 119883119883119883120601119883119883119872 minus11990311990310055761005576120572120572=120572

120598120598120572120572120578120578120572120572 119883119883119883119872 120578120578120572120572 11988311988311988311987210077031007703 120601 (37)

and using (32)

1198611198612 119883119883119883120601 119883119883119872 = minus120578120578 119883119883119883119872 119906119906 119883119883119883119872 minus 120578120578 119883119883119883119872 119906119906 119883119883119883119872 120601 (38)

for any119883119883120601119883119883 119883 119883119883119883119883119872119872119872 If 119867119867 = 120582 that is 119861119861120572 = 120582 then lightlikehypersurface119872119872 is said to be totally contact geodesic

Let 119877119877 be the curvature tensor elds of 119872119872119872 then for anindenite 119982119982-space form119872119872119883119872119872119872 we have (see [8])

119877119877 119883119883119883120601119883119883120601119883119883120601119883119883119872

=119872119872 + 1198881205981205984

10076821007682119892119892 10076501007650120601120601119883119883120601 12060112060111988311988310076661007666 119892119892 10076501007650120601120601119883119883120601 12060112060111988311988310076661007666

minus119892119892 10076501007650120601120601119883119883120601 12060112060111988311988310076661007666 119892119892 10076501007650120601120601119883119883120601 1206011206011198831198831007666100766610076981007698

+119872119872 minus 1205981205984

Φ 119883119883119883120601119883119883119872Φ 119883119883119883120601 119883119883119872 minus Φ 119883119883119883120601119883119883119872Φ 119883119883119883120601119883119883119872

+2Φ 119883119883119883120601119883119883119872Φ 119883119883119883120601119883119883119872

+ 10076821007682120578120578 119883119883119883119872 120578120578 119883119883119883119872 119892119892 10076501007650120601120601119883119883120601 12060112060111988311988310076661007666 minus 120578120578 119883119883119883119872 120578120578 119883119883119883119872 119892119892 10076501007650120601120601119883119883120601 12060112060111988311988310076661007666

+ 120578120578 119883119883119883119872 120578120578 119883119883119883119872 119892119892 10076501007650120601120601119883119883120601 12060112060111988311988310076661007666

minus120578120578 119883119883119883119872 120578120578 119883119883119883119872 119892119892 10076501007650120601120601119883119883120601 1206011206011198831198831007666100766610076981007698 120601(39)

for any119883119883120601119883119883120601119883119883120601119883119883 119883 119883119883119883119883119872119872119872 whereΦ119883119883119883120601119883119883119872 = 119892119892119883119883119883120601 120601120601119883119883119872 and120598120598 = 120598119903119903

120572120572=120572 120598120598120572120572 Using (39) we obtain

119877119877 119883119883119883120601119883119883120601119883119883120601 120585120585119872 =119872119872 minus 1205981205984

10076821007682119892119892 119883119883119883120601119883119883119872 119892119892 10076501007650119883119883120601 12060112060111988311988310076661007666

minus 119892119892 119883119883119883120601 119883119883119872 119892119892 10076501007650119883119883120601 12060112060111988311988310076661007666

+2119892119892 119883119883119883120601119883119883119872 119892119892 10076501007650119883119883120601 1206011206011198831198831007666100766610076981007698 119873

(40)

Also for the pair 120585120585120601119873119873 on119984119984 119984 119872119872 from (38) of page no94 of [3] we have

119877119877 119883119883119883120601119883119883120601119883119883120601 120585120585119872 = 10076491007649nabla11988311988311986111986110076651007665 119883119883119883120601119883119883119872 minus 10076491007649nabla11988311988311986111986110076651007665 119883119883119883120601119883119883119872

+ 120591120591 119883119883119883119872 119861119861 119883119883119883120601119883119883119872 minus 120591120591 119883119883119883119872 119861119861 119883119883119883120601119883119883119872 120601(41)

119883119883120601119883119883120601119883119883 119883 119883119883119883119883119872119872119872 and 120585120585 119883 119883119883119883119883119872119872⟂119872 where

10076491007649nabla11988311988311986111986110076651007665 119883119883119883120601119883119883119872 = 119883119883 119883 119861119861 119883119883119883120601119883119883119872 minus 119861119861 10076491007649nabla11988311988311988311988312060111988311988310076651007665

minus 119861119861 10076491007649119883119883120601 nabla11988311988311988311988310076651007665 119873(42)

Journal of Mathematics 5

Hence using (40) and (41) we obtain

10076491007649nabla11988311988311986111986110076651007665 1007665119884119884119884119884119884) minus 10076491007649nabla11988411988411986111986110076651007665 1007665119883119883119884119884119884)

=119888119888 minus 1198881198884

10076821007682119892119892 1007665119881119881119884119883119883) 119892119892 10076501007650119884119884119884 12060112060111988411988410076661007666 minus 119892119892 1007665119881119881119884 119884119884) 119892119892 10076501007650119884119884119884 12060112060111988311988310076661007666

+2119892119892 1007665119881119881119884119884119884) 119892119892 10076501007650119883119883119884 1206011206011198841198841007666100766610076661007666

minus 120591120591 1007665119883119883) 119861119861 1007665119884119884119884119884119884) + 120591120591 1007665119884119884) 119861119861 1007665119883119883119884119884119884) 119884

(43)

for any119883119883119884119884119884119884119884119884 119883 1198831007665119883119883119872119872)

Lemma 5 Let 1007665119872119872119884 119892119892) be a totally contact umbilical lightlikehypersurface of an indenite 119982119982-manifold 1007665119872119872119884 119892119892) then for any119883119883119884119884119884119884119884119884 119883 1198831007665119883119883119872119872) one has

10076491007649nabla119883119883119861119861110076651007665 1007665119884119884119884119884119884) = 1007665119883119883 119883 119883119883)10076871007687119892119892 1007665119884119884119884119884119884) minus11990311990310055761005576120572120572=1

119888119888120572120572120578120578120572120572 1007665119884119884) 120578120578120572120572 1007665119884119884)10077031007703

+ 119883119883 119861119861 1007665119883119883119884 119884119884) 120579120579 1007665119884119884) + 119861119861 1007665119883119883119884119884119884) 120579120579 1007665119884119884)

+ 119883119883120578120578 1007665119884119884) 10076811007681119892119892 1007649100764912060112060111988311988311988411988411988410076651007665 + 119906119906 1007665119883119883) 120579120579 1007665119884119884)10076971007697

+ 119883119883120578120578 1007665119884119884) 10076811007681119892119892 1007649100764912060112060111988311988311988411988411988410076651007665 + 119906119906 1007665119883119883) 120579120579 1007665119884119884)10076971007697

10076491007649nabla119883119883119861119861210076651007665 1007665119884119884119884119884119884) = 119888119888119906119906 1007665119884119884) 10076811007681119892119892 10076491007649119884119884119884 12060112060111988311988310076651007665 + 119906119906 1007665119883119883) 120579120579 1007665119884119884)10076971007697

+ 119888119888119906119906 1007665119884119884) 10076811007681119892119892 10076491007649119884119884119884 12060112060111988311988310076651007665 + 119906119906 1007665119883119883) 120579120579 1007665119884119884)10076971007697

+ 120578120578 1007665119884119884) 10076811007681119861119861 10076491007649119883119883119884 12060112060111988411988410076651007665 + 120591120591 1007665119883119883) 119906119906 1007665119884119884)10076971007697

+ 120578120578 1007665119884119884) 10076811007681119861119861 10076491007649119883119883119884 12060112060111988411988410076651007665 + 120591120591 1007665119883119883) 119906119906 1007665119884119884)10076971007697 (44)

Proof By virtue of (37) and (42) we obtain

10076491007649nabla119883119883119861119861110076651007665 1007665119884119884119884119884119884) = 1007665119883119883 119883 119883119883)10076871007687119892119892 1007665119884119884119884119884119884) minus11990311990310055761005576120572120572=1

119888119888120572120572120578120578120572120572 1007665119884119884) 120578120578120572120572 1007665119884119884)10077031007703

minus 119883119883 10076811007681119892119892 10076491007649nabla11988311988311988411988411988411988411988410076651007665 + 119892119892 10076491007649119884119884119884 nabla1198831198831198841198841007665100766510076971007697

+ 11988311988311990311990310055761005576120572120572=1

119888119888120572120572 10076811007681120578120578120572120572 10076491007649nabla11988311988311988411988410076651007665 120578120578

120572120572 1007665119884119884)

+120578120578120572120572 10076491007649nabla11988311988311988411988410076651007665 120578120578120572120572 1007665119884119884)10076971007697 (45)

Now using (12) in (45) we have

10076491007649nabla119883119883119861119861110076651007665 1007665119884119884119884119884119884) = 1007665119883119883 119883 119883119883)10076871007687119892119892 1007665119884119884119884119884119884) minus11990311990310055761005576120572120572=1

119888119888120572120572120578120578120572120572 1007665119884119884) 120578120578120572120572 1007665119884119884)10077031007703

minus 119883119883 10076811007681119883119883 10076491007649119892119892 1007665119884119884119884119884119884)10076651007665 minus 119861119861 1007665119883119883119884 119884119884) 120579120579 1007665119884119884)

minus119861119861 1007665119883119883119884119884119884) 120579120579 1007665119884119884)

+ 11988311988311990311990310055761005576120572120572=1

10076821007682119892119892 10076501007650nabla119883119883119884119884119884 12057712057712057212057210076661007666 120578120578120572120572 1007665119884119884)

+119892119892 10076501007650nabla119883119883119884119884119884 12057712057712057212057210076661007666 120578120578120572120572 1007665119884119884)10076661007666

(46)

Again using (12) we have

119892119892 10076501007650nabla119883119883119884119884119884 12057712057712057212057210076661007666 = 119883119883 10076501007650119892119892 10076501007650119884119884119884 1205771205771205721205721007666100766610076661007666 + 119888119888120572120572119892119892 10076491007649119884119884119884 12060112060111988311988310076651007665

+ 119888119888120572120572119906119906 1007665119883119883) 120579120579 1007665119884119884) (47)

us from (46) and (47) the rst expression of the theoremfollows Next using (38) and (42) we obtain

10076491007649nabla119883119883119861119861210076651007665 1007665119884119884119884119884119884) =11990311990310055761005576120572120572=1

119888119888120572120572 10076811007681120578120578120572120572 10076491007649nabla11988311988311988411988410076651007665 119906119906 1007665119884119884) + 120578120578120572120572 10076491007649nabla11988311988311988411988410076651007665119906119906 1007665119884119884)

+ 120578120578120572120572 1007665119884119884) 119906119906 10076491007649nabla11988311988311988411988410076651007665

+120578120578120572120572 1007665119884119884) 119906119906 10076491007649nabla1198831198831198841198841007665100766510076971007697 (48)

Using (31) and (32) we get

120578120578120572120572 10076491007649nabla11988311988311988411988410076651007665 = 119892119892 10076491007649119884119884119884 12060112060111988311988310076651007665 + 119906119906 1007665119883119883) 120579120579 1007665119884119884) 119884 (49)

and from (2) (3) (10) (16) (17) (18) and (25) we obtain

119906119906 10076491007649nabla11988311988311988411988410076651007665 = 119861119861 10076491007649119883119883119884 12060112060111988411988410076651007665 + 120591120591 1007665119883119883) 119906119906 1007665119884119884) (50)

us by substituting (49) and (50) in (48) we obtain (44)

eorem 6 Let 1198721198721007665119888119888) be an indenite 119982119982-space form and 119872119872be a totally contact umbilical lightlike hypersurface of 1198721198721007665119888119888)en 119888119888 = minus119888119888119888 where 119888119888 = 120598119903119903

120572120572=1 119888119888120572120572 and 119883119883 satises the followingdifferential equations

120585120585 119883 119883119883 + 119883119883120591120591 1007665120585120585) minus 1198831198832 = 0119884

119875119875119883119883 119883 119883119883 + 119883119883120591120591 1007665119875119875119883119883) = 0119884(51)

for any119883119883 119883 1198831007665119883119883119872119872)

Proof Let 119872119872 be a totally contact umbilical lightlike hyper-surface of an indenite 119982119982-space form 1198721198721007665119888119888) of constant 120601120601-sectional curvature 119888119888 en using (35) we have 1198611198611007665119883119883119884 120601120601119884119884) =1198831198831198921198921007665119883119883119884 120601120601119884119884) Substituting (29) (44) in (43) we obtain

1007665119883119883 119883 119883119883)10076871007687119892119892 1007665119884119884119884119884119884) minus11990311990310055761005576120572120572=1

119888119888120572120572120578120578120572120572 1007665119884119884) 120578120578120572120572 1007665119884119884)10077031007703

minus 1007665119884119884 119883 119883119883)10076871007687119892119892 1007665119883119883119884119884119884) minus11990311990310055761005576120572120572=1

119888119888120572120572120578120578120572120572 1007665119883119883) 120578120578120572120572 1007665119884119884)10077031007703

+ 119883119883 119861119861 1007665119883119883119884119884119884) 120579120579 1007665119884119884) minus 119861119861 1007665119884119884119884119884119884) 120579120579 1007665119883119883)

+ 2119883119883 10076811007681119892119892 1007649100764912060112060111988311988311988411988411988410076651007665 + 119906119906 1007665119883119883) 120579120579 1007665119884119884)10076971007697 120578120578 1007665119884119884)

+ 119883119883 1007681100768110076491007649119892119892 10076491007649119884119884119884 12060112060111988311988310076651007665 120578120578 1007665119884119884) minus 119892119892 10076491007649119884119884119884 12060112060111988411988410076651007665 120578120578 1007665119883119883)10076971007697

+ 119883119883 10076811007681119906119906 1007665119883119883) 120578120578 1007665119884119884) minus 119906119906 1007665119884119884) 120578120578 1007665119883119883)10076971007697 120579120579 1007665119884119884)

+ 2119888119888 10076811007681119892119892 1007649100764912060112060111988311988311988411988411988410076651007665 + 119906119906 1007665119883119883) 120579120579 1007665119884119884)10076971007697 119906119906 1007665119884119884)

+ 10076811007681119883119883119892119892 10076491007649119883119883119884 12060112060111988411988410076651007665 + 120591120591 1007665119883119883) 119906119906 1007665119884119884)

minus119883119883119892119892 10076491007649119884119884119884 12060112060111988311988310076651007665 minus 120591120591 1007665119884119884) 119906119906 1007665119883119883)10076971007697 120578120578 1007665119884119884)

6 Journal of Mathematics

+ 120582120582 10076811007681119892119892 10076491007649119883119883119883 11988311988311988311988310076651007665 120578120578 (119884119884) minus 119892119892 10076491007649119884119884119883 11988311988311988311988310076651007665 120578120578 (119883119883)10076971007697

+ 10076811007681120591120591 (119883119883) 120578120578 (119884119884) minus 120591120591 (119884119884) 120578120578 (119883119883)10076971007697 119906119906 (119883119883)

+ 120598120598 10076811007681119892119892 10076491007649119883119883119883 11988311988311988311988310076651007665 119906119906 (119884119884) minus 119892119892 10076491007649119883119883119883 11988311988311988411988410076651007665 119906119906 (119883119883)10076971007697

=119888119888 minus 1205981205984

10076821007682119892119892 (119881119881119883119883119883) 119892119892 10076501007650119883119883119883 11988311988311988411988410076661007666

minus 119892119892 (119881119881119883 119884119884) 119892119892 10076501007650119883119883119883 11988311988311988311988310076661007666

+2119892119892 (119881119881119883119883119883) 119892119892 10076501007650119883119883119883 1198831198831198841198841007666100766610076981007698

+ 120591120591 (119884119884) 119861119861 (119883119883119883119883119883) minus 120591120591 (119883119883) 119861119861 (119884119884119883119883119883) 119883(52)

for any119883119883119883119884119884119883119883119883 119883 119883(119883119883119872119872) Put119883119883 = 119883119883 in (52) we get

(119883119883 120585 120582120582) 10076871007687119892119892 (119884119884119883119883119883) minus11990311990310055761005576120572120572=120572

120598120598120572120572120578120578120572120572 (119884119884) 120578120578120572120572 (119883119883)10077031007703

minus 120582120582119861119861 (119884119884119883119883119883) minus 2120582120582119906119906 (119884119884) 120578120578 (119883119883)

minus 120582120582119906119906 (119883119883) 120578120578 (119884119884) minus 2120598120598119906119906 (119884119884) 119906119906 (119883119883)

+ 120591120591 (119883119883) 119906119906 (119884119884) 120578120578 (119883119883) + 2120582120582119906119906 (119884119884) 120578120578 (119883119883)

+ 120591120591 (119883119883) 119906119906 (119883119883) 120578120578 (119884119884) minus 120598120598119906119906 (119883119883) 119906119906 (119884119884)

+ 120582120582119906119906 (119883119883) 120578120578 (119884119884)

=119888119888 minus 1205981205984

119906119906 (119884119884) 119906119906 (119883119883) + 2119906119906 (119883119883) 119906119906 (119884119884)

minus 120591120591 (119883119883) 119861119861 (119884119884119883119883119883)

(53)

Put 119884119884 = 119883119883 = 119884119884 in (53) and then using 119892119892(119884119884119883119884119884) = 119892 119906119906(119884119884) =120572 120578120578120572120572(119884119884) = 119892 we obtain

minus120582120582119861119861 (119884119884119883119884119884) minus 3120598120598 = 3(119888119888 minus 120598120598)4

minus 120591120591 (119883119883) 119861119861 (119884119884119883119884119884) 119883 (54)

since 119861119861(119884119884119883119884119884) = 119892 we get 119888119888 = minus3120598120598 Moreover by putting 119884119884 =119881119881 and119883119883 = 119884119884 in (52) we obtain

119883119883 120585 120582120582 + 120582120582120591120591 (119883119883) minus 1205821205822 = 119892 (55)

Finally putting119883119883 = 119883119883119883119883 119884119884 = 119883119883119884119884 and119883119883 = 119883119883119883119883 in (52) with119888119888 = minus3120598120598 and using that 119878119878(119883119883119872119872) is nondegenerate we obtain

119883119883119883119883 120585 120582120582 + 120582120582120591120591 (119883119883119883119883)10076551007655119883119883119884119884 minus11990311990310055761005576120572120572=120572

120578120578120572120572 (119883119883119884119884) 12057712057712057212057210076711007671

= 119883119883119884119884 120585 120582120582 + 120582120582120591120591 (119883119883119884119884)10076551007655119883119883119883119883 minus11990311990310055761005576120572120572=120572

120578120578120572120572 (119883119883119883119883) 12057712057712057212057210076711007671

(56)

Putting 119883119883119883119883 = 120577120577120572120572 in (56) we get

10076821007682120577120577120572120572 120585 120582120582 + 120582120582120591120591 10076501007650120577120577120572120572100766610076661007698100769810076551007655119883119883119884119884 minus11990311990310055761005576120572120572=120572

120578120578120572120572 (119883119883119884119884) 12057712057712057212057210076711007671 = 119892 (57)

By taking 119884119884 = 119881119881 we obtain

120577120577120572120572 120585 120582120582 + 120582120582120591120591 1007650100765012057712057712057212057210076661007666 = 119892 (58)

Since 119883119883119883119883 119883 119883(119878119878(119883119883119872119872)) therefore using (19) we can write

119883119883119883119883 = 119906119906 (119883119883119883119883)119884119884 + 119880119880 (119883119883119883119883)119881119881 +2119899119899minus410055761005576119894119894=120572

120573120573119894119894119865119865119894119894

+11990311990310055761005576120572120572=120572

120598120598120572120572120578120578120572120572 (119883119883119883119883) 120577120577120572120572

= 119883119883119883119883prime +11990311990310055761005576120572120572=120572

120598120598120572120572120578120578120572120572 (119883119883119883119883) 120577120577120572120572119883

(59)

where 119865119865119894119894120572le119894119894le2119899119899minus4 is an orthogonal basis of 119863119863119892 then using(58) we have

119883119883119883119883 120585 120582120582 + 120582120582120591120591 (119883119883119883119883) = 10076551007655119883119883119883119883prime +11990311990310055761005576120572120572=120572

120598120598120572120572120578120578120572120572 (119883119883119883119883) 12057712057712057212057210076711007671 120585 120582120582

+ 12058212058212059112059110076551007655119883119883119883119883prime +11990311990310055761005576120572120572=120572

120598120598120572120572120578120578120572120572 (119883119883119883119883) 12057712057712057212057210076711007671

= 119883119883119883119883prime 120585 120582120582 + 120582120582120591120591 10076511007651119883119883119883119883prime10076671007667 119883

(60)

which leads to get from (56)

10076831007683119883119883119883119883prime 120585 120582120582 + 120582120582120591120591 10076511007651119883119883119883119883prime1007667100766710076991007699119883119883119884119884prime = 10076831007683119883119883119884119884prime 120585 120582120582 + 120582120582120591120591 10076511007651119883119883119884119884prime1007667100766710076991007699119883119883119883119883prime(61)

Now assume that there eists a vector eld 119883119883119892 on someneighborhood of119872119872 such that 119883119883119883119883prime

119892 120585 120582120582 + 120582120582120591120591(119883119883119883119883prime119892) ne 119892 at some

point 119901119901 in the neighborhood en from (61) it is clear thatall the vectors of the ber (119878119878(119883119883119872119872) minus 120577120577120572120572)119901119901 are collinear with(119883119883119883119883prime

119892)119901119901is contradicts dim(119878119878(119883119883119872119872)minus120577120577120572120572) gt 120572is impliesthe result

From the previous mentioned theorem we have thefollowing corollary

Corollary 7 ere exist no totally contact umbilical lightlikereal hypersurfaces of indenite 119982119982-space form119872119872(119888119888) (119888119888 ne minus 3120598120598)with 120577120577120572120572 119883 119883119883119872119872

Acknowledgment

e authors would like to thank the anonymous referee forhisher valuable suggestions that helped them to improve thispaper

References

[1] B Y Chen Geometry of Submanifolds Marcel Dekker NewYork USA 1973

[2] B OrsquoNeill Semi-Riemannian Geometry with Applications toRelativity Academic Press 1983

[3] K L Duggal and A Bejancu Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications Kluwer AcademicPublishers Dordrecht e Netherlands 1996

Journal of Mathematics 7

[4] L Duggal and B Sahin ldquoLightlike submanifolds of indeniteSasakian manifoldsrdquo International Journal of Mathematics andMathematical Sciences vol 2007 Article ID 57585 21 pages2007

[5] V I Arnold ldquoContact geometry the geometrical method ofGibbss thermodynamicsrdquo in Proceedings of the Gibbs Sym-posium pp 163ndash179 American Mathematical Society NewHaven Conn USA 1989

[6] SMaclaneGeometricalMechanics II Lecture Notes Universityof Chicago Chicago Ill USA 1968

[7] V E Nazaikinskii V E Shatalov and B Y Sternin ContactGeometry and Linear Differential Equations vol 6 of DeGruyter Expositions in Mathematics Walter de Gruyter BerlinGermany 1992

[8] L Brunetti and AM Pastore ldquoCurvature of a class of indeniteglobally framed f-manifoldsrdquo Bulletin Mathematique de laSociete des Sciences Mathematiques de Roumanie vol 51 no 3pp 183ndash204 2008

[9] C Calin ldquoOn existence of degenerate hypersurfaces in Sasakianmanifoldsrdquo Arab Journal of Mathematical Sciences vol 5 pp21ndash27 1999

[10] L Brunetti and A M Pastore ldquoLightlike hypersurfaces in inde-ite 119982119982-manifoldsrdquo Differential GeometrymdashDynamical Systemsvol 12 pp 18ndash40 2010

[11] A Bejancu ldquoUmbilical semi-invariant submanifolds of aSasakian manifoldrdquo Tensor vol 37 pp 203ndash213 1982

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2 Journal of Mathematics

vanishes where 119873119873120601120601 is Nijenhuis tensor of 120601120601 and 119889119889120578120578120572120572 = Φfor any 120572120572 120572 120572120572120572120572 120572 120572120572120572 where Φ(119883119883120572119883119883119883 = 119892119892(119883119883120572 120601120601119883119883119883 for any119883119883120572119883119883 120572 119883(119883119883119872119872119883 e Levi-Civita connection of an indenite119982119982-manifold satises

10076501007650nabla11988311988312060112060110076661007666119883119883 = 119892119892 10076501007650120601120601119883119883120572 12060112060111988311988310076661007666 120577120577 120577 120578120578 (119883119883119883 1206011206012(119883119883119883 120572 (3)

where 120577120577 = 120577120572120572120572120572=120572 120577120577120572120572 and 120578120578 = 120577

120572120572120572120572=120572 120598120598120572120572120578120578

120572120572 Also

nabla119883119883120577120577120572120572 = minus120598120598120572120572120601120601119883119883120572 (4)

and ker120601120601 is integrable at distribution since nabla120577120577120572120572120577120577120573120573 = 0 fordetail see [8]

We recall notations and fundamental equations for light-like hypersurfaces which are due to Duggal and Bejancu [3]

Let (119872119872120572 119892119892119883 be a (2119899119899 120577 120572119883-dimensional semi-Riemannianmanifold with index 119904119904120572 0 119904 119904119904 119904 2119899119899 120577 120572 and let (119872119872120572 119892119892119883be a hypersurface of 119872119872 with 119892119892 = 119892119892119892119872119872 119872119872 is a lightlikehypersurface of 119872119872 if 119892119892 is of constant rank 2119899119899 minus 120572 and thenormal bundle 119883119883119872119872⟂ is a distribution of rank 120572 on 119872119872 Acomplementary bundle of 119883119883119872119872⟂ in 119883119883119872119872 is a rank 2119899119899 minus 120572nondegenerate distribution over119872119872 It is denoted by 119878119878(119883119883119872119872119883and known as a screen distribution erefore we have

119883119883119872119872 = 119878119878 (119883119883119872119872119883 ⟂ 119883119883119872119872⟂120572

11988311988311987211987210092501009250119872119872 = 119878119878 (119883119883119872119872119883 ⟂ 119878119878(119883119883119872119872119883⟂120572(5)

where 119878119878(119883119883119872119872119883⟂ is a orthogonal complementary vector bundleof 119878119878(119883119883119872119872119883 in 119883119883119872119872119892119872119872 e following theorem has importantroles in studying the geometry of lightlike hypersurface

eorem1 Let (119872119872120572 119892119892120572 119878119878(119883119883119872119872119883119883 be a lightlike hypersurface of asemi-Riemannian manifold (119872119872120572 119892119892119883 en there exists a uniquevector bundle tr(119883119883119872119872119883 of rank 120572 over 119872119872 such that for anynonzero section 120585120585 of 119883119883119872119872⟂ on a coordinate neighborhood119984119984 119984119872119872 there exists a unique section119873119873 of tr(119883119883119872119872119883 on119984119984 satisfying

119892119892 (120585120585120572119873119873119883 = 120572120572 119892119892 (119873119873120572119873119873119883 = 119892119892 (119873119873120572119873119873119883 = 0120572

forall119873119873 120572 119883 10076491007649119878119878 (11988311988311987211987211988311989211998411998410076651007665 (6)

Hence we have the following decompositions of 119883119883119872119872119892119872119872

11988311988311987211987210092501009250119872119872 = 119878119878 (119883119883119872119872119883 ⟂ 10076501007650119883119883119872119872⟂ oplus tr (11988311988311987211987211988310076661007666

= 119883119883119872119872 oplus tr (119883119883119872119872119883 (7)

Let nabla be the Levi-Civita connection on 119872119872 with respectto 119892119892 en using the decompositions in (7) Gauss andWeingarten formulae are given as

nabla119883119883119883119883 = nabla119883119883119883119883 120577 119884 (119883119883120572 119883119883119883 120572

nabla119883119883119873119873 = minus119873119873119873119873119883119883 120577 nabla119905119905119883119883119873119873120572(8)

for any 119883119883120572119883119883 120572 119883(119883119883119872119872119883 and 119873119873 120572 119883(tr(119883119883119872119872119883119883 where nabla119883119883119883119883and 119873119873119873119873119883119883 belong to 119883(119883119883119872119872119883 while 119884(119883119883120572 119883119883119883 and nabla119905119905119883119883119873119873 belongto 119883(tr(119883119883119872119872119883119883 Here nabla is a torsion free linear connection

on 119872119872 119884 is a 119883(tr(119883119883119872119872119883119883-valued symmetric bilinear form on119883(119883119883119872119872119883 and known as the second fundamental form 119873119873119873119873 is alinear operator on 119883(119883119883119872119872119883 and known as the shape operatorof lightlike immersion and nabla119905119905 is a linear connection on119883(tr(119883119883119872119872119883119883

Locally for the pair 120572120585120585120572119873119873120572 following the Duggal andBejancursquos notation [3] we recall local second fundamentalform 119861119861 and 120572-form 120591120591 as

119861119861 (119883119883120572119883119883119883 = 119892119892 (119884 (119883119883120572 119883119883119883 120572 120585120585119883 120572

therefore 119884 (119883119883120572 119883119883119883 = 119861119861 (119883119883120572 119883119883119883119873119873

120591120591 (119883119883119883 = 119892119892 10076501007650nabla119905119905119883119883119873119873120572 12058512058510076661007666

therefore nabla119905119905119883119883119873119873 = 120591120591 (119883119883119883119873119873

(9)

Hence locally (8) becomes

nabla119883119883119883119883 = nabla119883119883119883119883 120577 119861119861 (119883119883120572119883119883119883119873119873120572

nabla119883119883119873119873 = minus119873119873119873119873119883119883 120577 120591120591 (119883119883119883119873119873(10)

If 119875119875 denotes the projection morphism of 119883119883119872119872 on 119878119878(119883119883119872119872119883then from (5) we have

nabla119883119883119875119875119883119883 = nablalowast119883119883119875119875119883119883 120577 119884lowast (119883119883120572 119875119875119883119883119883 120572

nabla119883119883120585120585 = minus119873119873lowast120585120585119883119883 120577 nablalowast119905119905119883119883 120585120585120572

(11)

for any 119883119883120572119883119883 120572 119883(119883119883119872119872119883 120585120585 120572 119883(119883119883119872119872⟂119883 where nablalowast119883119883119875119875119883119883and 119873119873lowast

120585120585119883119883 belong to 119883(119878119878(119883119883119872119872119883119883 while 119884lowast(119883119883120572 119875119875119883119883119883 and nablalowast119905119905119883119883 120585120585belong to 119883(119883119883119872119872⟂119883 Here 119884lowast and 119873119873lowast

120585120585 are called the secondfundamental form and the shape operator of the screendistribution respectively It should be noted that the inducedlinear connection nabla is not a metric connection as it satises

10076491007649nabla11988311988311989211989210076651007665 (119883119883120572119884119884119883 = 119861119861 (119883119883120572 119883119883119883 120579120579 (119884119884119883 120577 119861119861 (119883119883120572119884119884119883 120579120579 (119883119883119883 120572 (12)

where 120579120579 is a differential 120572 form locally dened on119872119872 by 120579120579(120579119883 =119892119892(119873119873120572 120579119883 By direct calculation we have

119892119892 10076491007649119873119873119873119873119883119883120572 11987511987511988311988310076651007665 = 119892119892 10076491007649119873119873120572 119884lowast (119883119883120572 11987511987511988311988311988310076651007665 120572 119892119892 1007649100764911987311987311987311987311988311988312057211987311987310076651007665 = 0120572

119892119892 10076501007650119873119873lowast120585120585119883119883120572 11987511987511988311988310076661007666 = 119892119892 (120585120585120572 119884 (119883119883120572 119875119875119883119883119883119883 120572 119892119892 10076501007650119873119873lowast

12058512058511988311988312057211987311987310076661007666 = 0120572(13)

for any 119883119883120572119883119883 120572 119883(119883119883119872119872119883 120585120585 120572 119883(119883119883119872119872⟂119883 and 119873119873 120572 119883(tr(119883119883119872119872119883119883According to Duggal and Bejancursquos notation [3] locally wehave know

119862119862 (119883119883120572 119875119875119883119883119883 = 119892119892 10076491007649119884lowast (119883119883120572 119875119875119883119883119883 12057211987311987310076651007665

therefore 119884lowast (119883119883120572 119875119875119883119883119883 = 119862119862 (119883119883120572 119875119875119883119883119883 120585120585120572

120598120598 (119883119883119883 = 119892119892 10076501007650nablalowast119905119905119883119883 12058512058512057211987311987310076661007666

therefore nablalowast119905119905119883119883 120585120585 = 120598120598 (119883119883119883 120585120585

(14)

en (11) become

nabla119883119883119875119875119883119883 = nablalowast119883119883119875119875119883119883 120577 119862119862 (119883119883120572 119875119875119883119883119883 120585120585120572 (15)

nabla119883119883120585120585 = minus119873119873lowast120585120585119883119883 120577 120598120598 (119883119883119883 120585120585 = minus119873119873lowast

120585120585119883119883 minus 120591120591 (119883119883119883 120585120585 (16)

Journal of Mathematics 3

and also give

119892119892 10076491007649119860119860119873119873119884119884119884 11988411988411988411988410076651007665 = 119862119862 (119884119884119884 119884119884119884119884) 119884 119892119892 1007649100764911986011986011987311987311988411988411988411987311987310076651007665 = 0119884

119892119892 10076501007650119860119860lowast120585120585119883119883119884 11988411988411988411988410076661007666 = 119861119861 (119883119883119884 119884119884119884119884) 119884 119892119892 10076501007650119860119860lowast

12058512058511988311988311988411987311987310076661007666 = 0(17)

From (9) it is clear that

119861119861 (119883119883119884 120585120585) = 0119884 (18)

that is the second fundamental form of a lightlikehypersurface is degenerate Following the notations ofDuggal and Bejancu [3] for the curvature tensor 119877119877 of119872119872 we have 119877119877(119883119883119884119884119884119884119877119877119884119884119884) = 119892119892(119877119877(119883119883119884 119884119884)119877119877119884119884119884) for any119883119883119884119884119884119884119877119877119884119884119884 119883 119883(119883119883119872119872)

3 Lightlike Hypersurface ofndenite119982119982-Manifolds

Let (119872119872119884 119892119892119884 119872119872(119883119883119872119872)) be a lightlike hypersurface of an indenite119892119892119892119892119892119892-manifold (1198721198722119899119899119899119899119899119884 120601120601119884 120577120577120572120572119884 120578120578

120572120572119884 119892119892) and 120572120572 119883 120572120572119884120572 119884 119899119899120572 suchthat the characteristic vector elds 120577120577120572120572 120572120572 119883 120572120572119884120572 119884 119899119899120572 aretangent to119872119872 As the ambient manifold119872119872 has an additionalgeometric structure 120601120601 we must look for a particular screendistribution on 119872119872 Since 119892119892(120601120601120585120585119884 120585120585) = 0 for any 120585120585 119883 119883(119883119883119872119872⟂)therefore 120601120601(119883119883119872119872⟂) is a distribution on119872119872 of rank 120572 such that119883119883119872119872⟂ cap120601120601(119883119883119872119872⟂) = 1205720120572 Moreover 119892119892(120601120601119873119873119884 120585120585) = 120601119892119892(119873119873119884 120601120601120585120585) = 0for any119873119873 119883 119883(119873119873(119883119883119872119872)) therefore 120601120601119873119873 is tangent to119872119872 Since119892119892(120601120601119873119873119884119873119873) = 0 that is the component of 120601120601119873119873 with respectto 120585120585 vanishes this implies 120601120601119873119873 119883 119883(119872119872(119883119883119872119872)) As 120585120585 and 119873119873 arenull vector elds satisfying 119892119892(120585120585119884119873119873) = 120572 therefore from (2)we deduce that 120601120601120585120585 and 120601120601119873119873 are null vector elds satisfying119892119892(120601120601120585120585119884 120601120601119873119873) = 120572 Hence 120601120601(119883119883119872119872⟂) oplus 120601120601(119873119873(119883119883119872119872)) is a vectorsubbundle of 119872119872(119883119883119872119872) of rank 2 It is known [9] that if structurevector elds 120577120577120572120572 120572120572 119883 120572120572120572 119884 119899119899120572 are tangent to119872119872 then 120577120577120572120572 belongto 119872119872(119883119883119872119872) erefore 119892119892(120601120601120585120585119884 120577120577120572120572) = 119892119892(120601120601119873119873119884 120577120577120572120572) = 0 implies thatthere exists a nondegenerate distribution 1198631198630 of rank 2119899119899 120601 119899on119872119872 such that

119872119872 (119883119883119872119872) = 10076821007682120601120601 (119873119873 (119883119883119872119872)) oplus 120601120601119883119883119872119872⟂10076981007698 ⟂ 1198631198630 ⟂ 1007682100768212057712057712057212057210076981007698 119884 (19)

therefore

119883119883119872119872 = 10076821007682120601120601 (119873119873 (119883119883119872119872)) oplus 120601120601119883119883119872119872⟂10076981007698 ⟂ 1198631198630 ⟂ 1007682100768212057712057712057212057210076981007698 ⟂ 119883119883119872119872⟂(20)

Let 119863119863 = 119883119883119872119872⟂ ⟂ 120601120601119883119883119872119872⟂ ⟂ 1198631198630 and let 119863119863119863 = 120601120601 119873119873(119883119883119872119872) then119863119863 clearly is invariant and119863119863119863 is anti-invariant under 120601120601 and wehave

119883119883119872119872 = 119863119863119863 oplus 119863119863 ⟂ 1007682100768212057712057712057212057210076981007698 (21)

onsider the local lightlike vector elds as

119880119880 = 120601120601120601119873119873 119883 120601120601 (119873119873 (119883119883119872119872)) = 119863119863119863119884

119881119881 = 120601120601120601120585120585 119883 119863119863(22)

Using the decompositions in (21) any 119883119883 119883 119883(119883119883119872119872) can bewritten as

119883119883 = 119872119872119883119883 119899 119883119883119883119883 11989911989911989910055761005576120572120572=120572

120578120578120572120572 (119883119883) 120577120577120572120572119884 (23)

where 119872119872 and 119883119883 are the projection morphisms of 119883119883119872119872 into 119863119863and119863119863119863 respectively andwhere119883119883119883119883 = 119876119876(119883119883)119880119880 and 119876119876 is a local120572-form on119872119872 dened by 119876119876(119883119883) = 119892119892(119881119881119884119883119883) erefore

119876119876 (119880119880) = 120572119884 119876119876 (119883119883) = 0119884 forall119883119883 119883 119883 (119863119863) 119884 1206011206012119873119873 = 120601119873119873 (24)

Applying 120601120601 to (23) and then using (24) we obtain 120601120601119883119883 =120601120601(119872119872119883119883) 119899 119876119876(119883119883)119873119873 By assuming 120601120601(119872119872119883119883) = 120601120601119883119883 for any 119883119883 119883119883(119883119883119872119872) we obtain a tensor eld 120601120601 of type (120572119884 120572) on 119872119872 andgiven by

120601120601119883119883 = 120601120601119883119883 119899 119876119876 (119883119883)119873119873 (25)

Applying 120601120601 to (25) and then using the denition of 120601120601119883119883 weobtain

1206011206012119883119883 = 120601119883119883 119899 119876119876 (119883119883)119880119880 11989911989911989910055761005576120572120572=120572

120578120578120572120572 (119883119883) 120577120577120572120572 (26)

Moreover since 120601120601119883119883 = 120601120601(119872119872119883119883) therefore

120601120601119880119880 = 0119884 120578120578120572120572 ∘ 120601120601 = 0119884 119876119876 1007649100764912060112060111988311988310076651007665 = 0119884 (27)

for any 119883119883 119883 119883(119883119883119872119872) us we have the following theoremanalogous to a theorem proved in [10]

eorem 2 Let (119872119872119884 120601120601119884 120577120577120572120572119884 120578120578120572120572119884 119892119892) be an indenite 119892119892119892119892119892119892-

manifold and let (119872119872119884 119892119892119884 119872119872(119883119883119872119872)) be a lightlike hypersurface of119872119872 then (119872119872119884 120601120601119884 120577120577120572120572119884 119880119880119884 120578120578

120572120572119884 119876119876) is also a 119892119892119892119892119892119892-manifold

Since 120601120601119880119880 = 0 therefore applying 120601120601 to (26) we get 1206011206013 119899120601120601 =0 this implies that 120601120601 is an 119892119892 structure of constant rank on119872119872Now using (2) and (25) we obtain

119892119892 10076491007649120601120601119883119883119884 12060112060111988411988410076651007665 = 119892119892 (119883119883119884 119884119884) 120601 119876119876 (119883119883) 120584120584 (119884119884)

120601 119876119876 (119884119884) 120584120584 (119883119883) 12060111989911989910055761005576120572120572=120572

120598120598120572120572120578120578120572120572 (119883119883) 120578120578120572120572 (119884119884) 119884

(28)

for any119883119883119884119884119884 119883 119883(119883119883119872119872) where 120584120584 is a 120572 form locally dened on119872119872 by 120584120584(120584) = 119892119892(119880119880119884 120584) Replace119883119883 by 120601120601119883119883 in (28) and using (26)and (27) we derive

119892119892 1007649100764912060112060111988311988311988411988411988410076651007665 119899 119892119892 10076491007649119883119883119884 12060112060111988411988410076651007665

= 120601119876119876 (119883119883) 120579120579 (119884119884) 120601 119876119876 (119884119884) 120579120579 (119883119883) (29)

Let 119872119872 be a lightlike hypersurface of an indenite 119982119982-manifold119872119872 then by using (4) (10) and (25) we obtain

nabla119883119883120577120577120572120572 119899 119861119861 10076501007650119883119883119884 12057712057712057212057210076661007666119873119873 = 120601120598120598120572120572120601120601119883119883 120601 120598120598120572120572119876119876 (119883119883)119873119873119884 (30)

4 Journal of Mathematics

for any 119883119883 119883 119883119883119883119883119872119872119872 then comparing the tangential andtransversal components we get

nabla119883119883120577120577120572120572 = minus120598120598120572120572120601120601119883119883120601 (31)

119861119861 10076501007650119883119883120601 12057712057712057212057210076661007666 = minus120598120598120572120572119906119906 119883119883119883119872 120601

this implies ℎ 10076501007650119883119883120601 12057712057712057212057210076661007666 = minus120598120598120572120572119906119906 119883119883119883119872119873119873119873(32)

Using (15) and (31) we get nablalowast119883119883120577120577120572120572 + 119862119862119883119883119883120601 120577120577120572120572119872120585120585 = minus120598120598120572120572120601120601119883119883taking inner product with119873119873 119883 119883119883119873119873119883119883119883119872119872119872119872 we get119862119862119883119883119883120601 120577120577120572120572119872 =minus120598120598120572120572119892119892119883120601120601119883119883120601119873119873119872 and then using (29) we obtain

11986211986210076501007650119883119883120601 12057712057712057212057210076661007666 = 120598120598120572120572119892119892 10076491007649119883119883120601 12060112060111987311987310076651007665 + 119906119906 119883119883119883119872 120579120579 119883119873119873119872 minus 119906119906 119883119873119873119872 120579120579 119883119883119883119872

= minus120598120598120572120572119892119892 119883119883119883120601119883119883119872

= minus120598120598120572120572120584120584 119883119883119883119872 119873

(33)

Using (3) for any119873119873 119883 119883119883119873119873119883119883119883119872119872119872119872 we get nabla119883119883120601120601119873119873 = 120601120601nabla119883119883119873119873 +119892119892119883119883119883120601119873119873119872120577120577 therefore

119861119861 119883119883119883120601119883119883119872 = 119892119892 119883ℎ 119883119883119883120601119883119883119872 120601 120585120585119872

= minus119892119892 10076501007650nabla119883119883120601120601119873119873120601 12058512058510076661007666

= 119892119892 10076501007650nabla119883119883120601 12060112060112058512058510076661007666

= 119892119892 1007649100764911986011986011987311987311988311988312060111988311988310076651007665

= 119862119862 119883119883119883120601119883119883119872 119873

(34)

4 Totally Contact Umbilical Lightlikeyesface of ndenite119982119982-Manifolds

ere wewill follow eancus denition [11] of totally contactumbilical submanifolds of Sasakian manifolds to state totallycontact umbilical lightlike hypersurfaces of indenite 119982119982-manifolds

e A lightlike hypersurface 119872119872 of an indenite119982119982-manifold is said to be totally contact umbilical lightlikehypersurface if the second fundamental form ℎ of119872119872 satises

ℎ 119883119883119883120601 119883119883119872 = 10076871007687119892119892 119883119883119883120601119883119883119872 minus11990311990310055761005576120572120572=120572

120598120598120572120572120578120578120572120572 119883119883119883119872 120578120578120572120572 11988311988311988311987210077031007703119867119867

+11990311990310055761005576120572120572=120572

120578120578120572120572 119883119883119883119872 ℎ 10076501007650119883119883120601 12057712057712057212057210076661007666

+11990311990310055761005576120572120572=120572

120578120578120572120572 119883119883119883119872 ℎ 10076501007650119883119883120601 12057712057712057212057210076661007666 120601

(35)

for any 119883119883120601119883119883 119883 119883119883119883119883119872119872119872 where119867119867 is a transversal vector eldon 119872119872 that is 119867119867 = 119867119867119873119873 where 119867119867 is a smooth function on119984119984 119984 119872119872

Remark 4 We can also write (35) asℎ 119883119883119883120601 119883119883119872 = 119861119861 119883119883119883120601 119883119883119872119873119873

= 10076811007681119861119861120572 119883119883119883120601 119883119883119872 + 1198611198612 119883119883119883120601 11988311988311987210076971007697119873119873120601(36)

where

119861119861120572 119883119883119883120601 119883119883119872 = 11986711986710076871007687119892119892 119883119883119883120601119883119883119872 minus11990311990310055761005576120572120572=120572

120598120598120572120572120578120578120572120572 119883119883119883119872 120578120578120572120572 11988311988311988311987210077031007703 120601 (37)

and using (32)

1198611198612 119883119883119883120601 119883119883119872 = minus120578120578 119883119883119883119872 119906119906 119883119883119883119872 minus 120578120578 119883119883119883119872 119906119906 119883119883119883119872 120601 (38)

for any119883119883120601119883119883 119883 119883119883119883119883119872119872119872 If 119867119867 = 120582 that is 119861119861120572 = 120582 then lightlikehypersurface119872119872 is said to be totally contact geodesic

Let 119877119877 be the curvature tensor elds of 119872119872119872 then for anindenite 119982119982-space form119872119872119883119872119872119872 we have (see [8])

119877119877 119883119883119883120601119883119883120601119883119883120601119883119883119872

=119872119872 + 1198881205981205984

10076821007682119892119892 10076501007650120601120601119883119883120601 12060112060111988311988310076661007666 119892119892 10076501007650120601120601119883119883120601 12060112060111988311988310076661007666

minus119892119892 10076501007650120601120601119883119883120601 12060112060111988311988310076661007666 119892119892 10076501007650120601120601119883119883120601 1206011206011198831198831007666100766610076981007698

+119872119872 minus 1205981205984

Φ 119883119883119883120601119883119883119872Φ 119883119883119883120601 119883119883119872 minus Φ 119883119883119883120601119883119883119872Φ 119883119883119883120601119883119883119872

+2Φ 119883119883119883120601119883119883119872Φ 119883119883119883120601119883119883119872

+ 10076821007682120578120578 119883119883119883119872 120578120578 119883119883119883119872 119892119892 10076501007650120601120601119883119883120601 12060112060111988311988310076661007666 minus 120578120578 119883119883119883119872 120578120578 119883119883119883119872 119892119892 10076501007650120601120601119883119883120601 12060112060111988311988310076661007666

+ 120578120578 119883119883119883119872 120578120578 119883119883119883119872 119892119892 10076501007650120601120601119883119883120601 12060112060111988311988310076661007666

minus120578120578 119883119883119883119872 120578120578 119883119883119883119872 119892119892 10076501007650120601120601119883119883120601 1206011206011198831198831007666100766610076981007698 120601(39)

for any119883119883120601119883119883120601119883119883120601119883119883 119883 119883119883119883119883119872119872119872 whereΦ119883119883119883120601119883119883119872 = 119892119892119883119883119883120601 120601120601119883119883119872 and120598120598 = 120598119903119903

120572120572=120572 120598120598120572120572 Using (39) we obtain

119877119877 119883119883119883120601119883119883120601119883119883120601 120585120585119872 =119872119872 minus 1205981205984

10076821007682119892119892 119883119883119883120601119883119883119872 119892119892 10076501007650119883119883120601 12060112060111988311988310076661007666

minus 119892119892 119883119883119883120601 119883119883119872 119892119892 10076501007650119883119883120601 12060112060111988311988310076661007666

+2119892119892 119883119883119883120601119883119883119872 119892119892 10076501007650119883119883120601 1206011206011198831198831007666100766610076981007698 119873

(40)

Also for the pair 120585120585120601119873119873 on119984119984 119984 119872119872 from (38) of page no94 of [3] we have

119877119877 119883119883119883120601119883119883120601119883119883120601 120585120585119872 = 10076491007649nabla11988311988311986111986110076651007665 119883119883119883120601119883119883119872 minus 10076491007649nabla11988311988311986111986110076651007665 119883119883119883120601119883119883119872

+ 120591120591 119883119883119883119872 119861119861 119883119883119883120601119883119883119872 minus 120591120591 119883119883119883119872 119861119861 119883119883119883120601119883119883119872 120601(41)

119883119883120601119883119883120601119883119883 119883 119883119883119883119883119872119872119872 and 120585120585 119883 119883119883119883119883119872119872⟂119872 where

10076491007649nabla11988311988311986111986110076651007665 119883119883119883120601119883119883119872 = 119883119883 119883 119861119861 119883119883119883120601119883119883119872 minus 119861119861 10076491007649nabla11988311988311988311988312060111988311988310076651007665

minus 119861119861 10076491007649119883119883120601 nabla11988311988311988311988310076651007665 119873(42)

Journal of Mathematics 5

Hence using (40) and (41) we obtain

10076491007649nabla11988311988311986111986110076651007665 1007665119884119884119884119884119884) minus 10076491007649nabla11988411988411986111986110076651007665 1007665119883119883119884119884119884)

=119888119888 minus 1198881198884

10076821007682119892119892 1007665119881119881119884119883119883) 119892119892 10076501007650119884119884119884 12060112060111988411988410076661007666 minus 119892119892 1007665119881119881119884 119884119884) 119892119892 10076501007650119884119884119884 12060112060111988311988310076661007666

+2119892119892 1007665119881119881119884119884119884) 119892119892 10076501007650119883119883119884 1206011206011198841198841007666100766610076661007666

minus 120591120591 1007665119883119883) 119861119861 1007665119884119884119884119884119884) + 120591120591 1007665119884119884) 119861119861 1007665119883119883119884119884119884) 119884

(43)

for any119883119883119884119884119884119884119884119884 119883 1198831007665119883119883119872119872)

Lemma 5 Let 1007665119872119872119884 119892119892) be a totally contact umbilical lightlikehypersurface of an indenite 119982119982-manifold 1007665119872119872119884 119892119892) then for any119883119883119884119884119884119884119884119884 119883 1198831007665119883119883119872119872) one has

10076491007649nabla119883119883119861119861110076651007665 1007665119884119884119884119884119884) = 1007665119883119883 119883 119883119883)10076871007687119892119892 1007665119884119884119884119884119884) minus11990311990310055761005576120572120572=1

119888119888120572120572120578120578120572120572 1007665119884119884) 120578120578120572120572 1007665119884119884)10077031007703

+ 119883119883 119861119861 1007665119883119883119884 119884119884) 120579120579 1007665119884119884) + 119861119861 1007665119883119883119884119884119884) 120579120579 1007665119884119884)

+ 119883119883120578120578 1007665119884119884) 10076811007681119892119892 1007649100764912060112060111988311988311988411988411988410076651007665 + 119906119906 1007665119883119883) 120579120579 1007665119884119884)10076971007697

+ 119883119883120578120578 1007665119884119884) 10076811007681119892119892 1007649100764912060112060111988311988311988411988411988410076651007665 + 119906119906 1007665119883119883) 120579120579 1007665119884119884)10076971007697

10076491007649nabla119883119883119861119861210076651007665 1007665119884119884119884119884119884) = 119888119888119906119906 1007665119884119884) 10076811007681119892119892 10076491007649119884119884119884 12060112060111988311988310076651007665 + 119906119906 1007665119883119883) 120579120579 1007665119884119884)10076971007697

+ 119888119888119906119906 1007665119884119884) 10076811007681119892119892 10076491007649119884119884119884 12060112060111988311988310076651007665 + 119906119906 1007665119883119883) 120579120579 1007665119884119884)10076971007697

+ 120578120578 1007665119884119884) 10076811007681119861119861 10076491007649119883119883119884 12060112060111988411988410076651007665 + 120591120591 1007665119883119883) 119906119906 1007665119884119884)10076971007697

+ 120578120578 1007665119884119884) 10076811007681119861119861 10076491007649119883119883119884 12060112060111988411988410076651007665 + 120591120591 1007665119883119883) 119906119906 1007665119884119884)10076971007697 (44)

Proof By virtue of (37) and (42) we obtain

10076491007649nabla119883119883119861119861110076651007665 1007665119884119884119884119884119884) = 1007665119883119883 119883 119883119883)10076871007687119892119892 1007665119884119884119884119884119884) minus11990311990310055761005576120572120572=1

119888119888120572120572120578120578120572120572 1007665119884119884) 120578120578120572120572 1007665119884119884)10077031007703

minus 119883119883 10076811007681119892119892 10076491007649nabla11988311988311988411988411988411988411988410076651007665 + 119892119892 10076491007649119884119884119884 nabla1198831198831198841198841007665100766510076971007697

+ 11988311988311990311990310055761005576120572120572=1

119888119888120572120572 10076811007681120578120578120572120572 10076491007649nabla11988311988311988411988410076651007665 120578120578

120572120572 1007665119884119884)

+120578120578120572120572 10076491007649nabla11988311988311988411988410076651007665 120578120578120572120572 1007665119884119884)10076971007697 (45)

Now using (12) in (45) we have

10076491007649nabla119883119883119861119861110076651007665 1007665119884119884119884119884119884) = 1007665119883119883 119883 119883119883)10076871007687119892119892 1007665119884119884119884119884119884) minus11990311990310055761005576120572120572=1

119888119888120572120572120578120578120572120572 1007665119884119884) 120578120578120572120572 1007665119884119884)10077031007703

minus 119883119883 10076811007681119883119883 10076491007649119892119892 1007665119884119884119884119884119884)10076651007665 minus 119861119861 1007665119883119883119884 119884119884) 120579120579 1007665119884119884)

minus119861119861 1007665119883119883119884119884119884) 120579120579 1007665119884119884)

+ 11988311988311990311990310055761005576120572120572=1

10076821007682119892119892 10076501007650nabla119883119883119884119884119884 12057712057712057212057210076661007666 120578120578120572120572 1007665119884119884)

+119892119892 10076501007650nabla119883119883119884119884119884 12057712057712057212057210076661007666 120578120578120572120572 1007665119884119884)10076661007666

(46)

Again using (12) we have

119892119892 10076501007650nabla119883119883119884119884119884 12057712057712057212057210076661007666 = 119883119883 10076501007650119892119892 10076501007650119884119884119884 1205771205771205721205721007666100766610076661007666 + 119888119888120572120572119892119892 10076491007649119884119884119884 12060112060111988311988310076651007665

+ 119888119888120572120572119906119906 1007665119883119883) 120579120579 1007665119884119884) (47)

us from (46) and (47) the rst expression of the theoremfollows Next using (38) and (42) we obtain

10076491007649nabla119883119883119861119861210076651007665 1007665119884119884119884119884119884) =11990311990310055761005576120572120572=1

119888119888120572120572 10076811007681120578120578120572120572 10076491007649nabla11988311988311988411988410076651007665 119906119906 1007665119884119884) + 120578120578120572120572 10076491007649nabla11988311988311988411988410076651007665119906119906 1007665119884119884)

+ 120578120578120572120572 1007665119884119884) 119906119906 10076491007649nabla11988311988311988411988410076651007665

+120578120578120572120572 1007665119884119884) 119906119906 10076491007649nabla1198831198831198841198841007665100766510076971007697 (48)

Using (31) and (32) we get

120578120578120572120572 10076491007649nabla11988311988311988411988410076651007665 = 119892119892 10076491007649119884119884119884 12060112060111988311988310076651007665 + 119906119906 1007665119883119883) 120579120579 1007665119884119884) 119884 (49)

and from (2) (3) (10) (16) (17) (18) and (25) we obtain

119906119906 10076491007649nabla11988311988311988411988410076651007665 = 119861119861 10076491007649119883119883119884 12060112060111988411988410076651007665 + 120591120591 1007665119883119883) 119906119906 1007665119884119884) (50)

us by substituting (49) and (50) in (48) we obtain (44)

eorem 6 Let 1198721198721007665119888119888) be an indenite 119982119982-space form and 119872119872be a totally contact umbilical lightlike hypersurface of 1198721198721007665119888119888)en 119888119888 = minus119888119888119888 where 119888119888 = 120598119903119903

120572120572=1 119888119888120572120572 and 119883119883 satises the followingdifferential equations

120585120585 119883 119883119883 + 119883119883120591120591 1007665120585120585) minus 1198831198832 = 0119884

119875119875119883119883 119883 119883119883 + 119883119883120591120591 1007665119875119875119883119883) = 0119884(51)

for any119883119883 119883 1198831007665119883119883119872119872)

Proof Let 119872119872 be a totally contact umbilical lightlike hyper-surface of an indenite 119982119982-space form 1198721198721007665119888119888) of constant 120601120601-sectional curvature 119888119888 en using (35) we have 1198611198611007665119883119883119884 120601120601119884119884) =1198831198831198921198921007665119883119883119884 120601120601119884119884) Substituting (29) (44) in (43) we obtain

1007665119883119883 119883 119883119883)10076871007687119892119892 1007665119884119884119884119884119884) minus11990311990310055761005576120572120572=1

119888119888120572120572120578120578120572120572 1007665119884119884) 120578120578120572120572 1007665119884119884)10077031007703

minus 1007665119884119884 119883 119883119883)10076871007687119892119892 1007665119883119883119884119884119884) minus11990311990310055761005576120572120572=1

119888119888120572120572120578120578120572120572 1007665119883119883) 120578120578120572120572 1007665119884119884)10077031007703

+ 119883119883 119861119861 1007665119883119883119884119884119884) 120579120579 1007665119884119884) minus 119861119861 1007665119884119884119884119884119884) 120579120579 1007665119883119883)

+ 2119883119883 10076811007681119892119892 1007649100764912060112060111988311988311988411988411988410076651007665 + 119906119906 1007665119883119883) 120579120579 1007665119884119884)10076971007697 120578120578 1007665119884119884)

+ 119883119883 1007681100768110076491007649119892119892 10076491007649119884119884119884 12060112060111988311988310076651007665 120578120578 1007665119884119884) minus 119892119892 10076491007649119884119884119884 12060112060111988411988410076651007665 120578120578 1007665119883119883)10076971007697

+ 119883119883 10076811007681119906119906 1007665119883119883) 120578120578 1007665119884119884) minus 119906119906 1007665119884119884) 120578120578 1007665119883119883)10076971007697 120579120579 1007665119884119884)

+ 2119888119888 10076811007681119892119892 1007649100764912060112060111988311988311988411988411988410076651007665 + 119906119906 1007665119883119883) 120579120579 1007665119884119884)10076971007697 119906119906 1007665119884119884)

+ 10076811007681119883119883119892119892 10076491007649119883119883119884 12060112060111988411988410076651007665 + 120591120591 1007665119883119883) 119906119906 1007665119884119884)

minus119883119883119892119892 10076491007649119884119884119884 12060112060111988311988310076651007665 minus 120591120591 1007665119884119884) 119906119906 1007665119883119883)10076971007697 120578120578 1007665119884119884)

6 Journal of Mathematics

+ 120582120582 10076811007681119892119892 10076491007649119883119883119883 11988311988311988311988310076651007665 120578120578 (119884119884) minus 119892119892 10076491007649119884119884119883 11988311988311988311988310076651007665 120578120578 (119883119883)10076971007697

+ 10076811007681120591120591 (119883119883) 120578120578 (119884119884) minus 120591120591 (119884119884) 120578120578 (119883119883)10076971007697 119906119906 (119883119883)

+ 120598120598 10076811007681119892119892 10076491007649119883119883119883 11988311988311988311988310076651007665 119906119906 (119884119884) minus 119892119892 10076491007649119883119883119883 11988311988311988411988410076651007665 119906119906 (119883119883)10076971007697

=119888119888 minus 1205981205984

10076821007682119892119892 (119881119881119883119883119883) 119892119892 10076501007650119883119883119883 11988311988311988411988410076661007666

minus 119892119892 (119881119881119883 119884119884) 119892119892 10076501007650119883119883119883 11988311988311988311988310076661007666

+2119892119892 (119881119881119883119883119883) 119892119892 10076501007650119883119883119883 1198831198831198841198841007666100766610076981007698

+ 120591120591 (119884119884) 119861119861 (119883119883119883119883119883) minus 120591120591 (119883119883) 119861119861 (119884119884119883119883119883) 119883(52)

for any119883119883119883119884119884119883119883119883 119883 119883(119883119883119872119872) Put119883119883 = 119883119883 in (52) we get

(119883119883 120585 120582120582) 10076871007687119892119892 (119884119884119883119883119883) minus11990311990310055761005576120572120572=120572

120598120598120572120572120578120578120572120572 (119884119884) 120578120578120572120572 (119883119883)10077031007703

minus 120582120582119861119861 (119884119884119883119883119883) minus 2120582120582119906119906 (119884119884) 120578120578 (119883119883)

minus 120582120582119906119906 (119883119883) 120578120578 (119884119884) minus 2120598120598119906119906 (119884119884) 119906119906 (119883119883)

+ 120591120591 (119883119883) 119906119906 (119884119884) 120578120578 (119883119883) + 2120582120582119906119906 (119884119884) 120578120578 (119883119883)

+ 120591120591 (119883119883) 119906119906 (119883119883) 120578120578 (119884119884) minus 120598120598119906119906 (119883119883) 119906119906 (119884119884)

+ 120582120582119906119906 (119883119883) 120578120578 (119884119884)

=119888119888 minus 1205981205984

119906119906 (119884119884) 119906119906 (119883119883) + 2119906119906 (119883119883) 119906119906 (119884119884)

minus 120591120591 (119883119883) 119861119861 (119884119884119883119883119883)

(53)

Put 119884119884 = 119883119883 = 119884119884 in (53) and then using 119892119892(119884119884119883119884119884) = 119892 119906119906(119884119884) =120572 120578120578120572120572(119884119884) = 119892 we obtain

minus120582120582119861119861 (119884119884119883119884119884) minus 3120598120598 = 3(119888119888 minus 120598120598)4

minus 120591120591 (119883119883) 119861119861 (119884119884119883119884119884) 119883 (54)

since 119861119861(119884119884119883119884119884) = 119892 we get 119888119888 = minus3120598120598 Moreover by putting 119884119884 =119881119881 and119883119883 = 119884119884 in (52) we obtain

119883119883 120585 120582120582 + 120582120582120591120591 (119883119883) minus 1205821205822 = 119892 (55)

Finally putting119883119883 = 119883119883119883119883 119884119884 = 119883119883119884119884 and119883119883 = 119883119883119883119883 in (52) with119888119888 = minus3120598120598 and using that 119878119878(119883119883119872119872) is nondegenerate we obtain

119883119883119883119883 120585 120582120582 + 120582120582120591120591 (119883119883119883119883)10076551007655119883119883119884119884 minus11990311990310055761005576120572120572=120572

120578120578120572120572 (119883119883119884119884) 12057712057712057212057210076711007671

= 119883119883119884119884 120585 120582120582 + 120582120582120591120591 (119883119883119884119884)10076551007655119883119883119883119883 minus11990311990310055761005576120572120572=120572

120578120578120572120572 (119883119883119883119883) 12057712057712057212057210076711007671

(56)

Putting 119883119883119883119883 = 120577120577120572120572 in (56) we get

10076821007682120577120577120572120572 120585 120582120582 + 120582120582120591120591 10076501007650120577120577120572120572100766610076661007698100769810076551007655119883119883119884119884 minus11990311990310055761005576120572120572=120572

120578120578120572120572 (119883119883119884119884) 12057712057712057212057210076711007671 = 119892 (57)

By taking 119884119884 = 119881119881 we obtain

120577120577120572120572 120585 120582120582 + 120582120582120591120591 1007650100765012057712057712057212057210076661007666 = 119892 (58)

Since 119883119883119883119883 119883 119883(119878119878(119883119883119872119872)) therefore using (19) we can write

119883119883119883119883 = 119906119906 (119883119883119883119883)119884119884 + 119880119880 (119883119883119883119883)119881119881 +2119899119899minus410055761005576119894119894=120572

120573120573119894119894119865119865119894119894

+11990311990310055761005576120572120572=120572

120598120598120572120572120578120578120572120572 (119883119883119883119883) 120577120577120572120572

= 119883119883119883119883prime +11990311990310055761005576120572120572=120572

120598120598120572120572120578120578120572120572 (119883119883119883119883) 120577120577120572120572119883

(59)

where 119865119865119894119894120572le119894119894le2119899119899minus4 is an orthogonal basis of 119863119863119892 then using(58) we have

119883119883119883119883 120585 120582120582 + 120582120582120591120591 (119883119883119883119883) = 10076551007655119883119883119883119883prime +11990311990310055761005576120572120572=120572

120598120598120572120572120578120578120572120572 (119883119883119883119883) 12057712057712057212057210076711007671 120585 120582120582

+ 12058212058212059112059110076551007655119883119883119883119883prime +11990311990310055761005576120572120572=120572

120598120598120572120572120578120578120572120572 (119883119883119883119883) 12057712057712057212057210076711007671

= 119883119883119883119883prime 120585 120582120582 + 120582120582120591120591 10076511007651119883119883119883119883prime10076671007667 119883

(60)

which leads to get from (56)

10076831007683119883119883119883119883prime 120585 120582120582 + 120582120582120591120591 10076511007651119883119883119883119883prime1007667100766710076991007699119883119883119884119884prime = 10076831007683119883119883119884119884prime 120585 120582120582 + 120582120582120591120591 10076511007651119883119883119884119884prime1007667100766710076991007699119883119883119883119883prime(61)

Now assume that there eists a vector eld 119883119883119892 on someneighborhood of119872119872 such that 119883119883119883119883prime

119892 120585 120582120582 + 120582120582120591120591(119883119883119883119883prime119892) ne 119892 at some

point 119901119901 in the neighborhood en from (61) it is clear thatall the vectors of the ber (119878119878(119883119883119872119872) minus 120577120577120572120572)119901119901 are collinear with(119883119883119883119883prime

119892)119901119901is contradicts dim(119878119878(119883119883119872119872)minus120577120577120572120572) gt 120572is impliesthe result

From the previous mentioned theorem we have thefollowing corollary

Corollary 7 ere exist no totally contact umbilical lightlikereal hypersurfaces of indenite 119982119982-space form119872119872(119888119888) (119888119888 ne minus 3120598120598)with 120577120577120572120572 119883 119883119883119872119872

Acknowledgment

e authors would like to thank the anonymous referee forhisher valuable suggestions that helped them to improve thispaper

References

[1] B Y Chen Geometry of Submanifolds Marcel Dekker NewYork USA 1973

[2] B OrsquoNeill Semi-Riemannian Geometry with Applications toRelativity Academic Press 1983

[3] K L Duggal and A Bejancu Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications Kluwer AcademicPublishers Dordrecht e Netherlands 1996

Journal of Mathematics 7

[4] L Duggal and B Sahin ldquoLightlike submanifolds of indeniteSasakian manifoldsrdquo International Journal of Mathematics andMathematical Sciences vol 2007 Article ID 57585 21 pages2007

[5] V I Arnold ldquoContact geometry the geometrical method ofGibbss thermodynamicsrdquo in Proceedings of the Gibbs Sym-posium pp 163ndash179 American Mathematical Society NewHaven Conn USA 1989

[6] SMaclaneGeometricalMechanics II Lecture Notes Universityof Chicago Chicago Ill USA 1968

[7] V E Nazaikinskii V E Shatalov and B Y Sternin ContactGeometry and Linear Differential Equations vol 6 of DeGruyter Expositions in Mathematics Walter de Gruyter BerlinGermany 1992

[8] L Brunetti and AM Pastore ldquoCurvature of a class of indeniteglobally framed f-manifoldsrdquo Bulletin Mathematique de laSociete des Sciences Mathematiques de Roumanie vol 51 no 3pp 183ndash204 2008

[9] C Calin ldquoOn existence of degenerate hypersurfaces in Sasakianmanifoldsrdquo Arab Journal of Mathematical Sciences vol 5 pp21ndash27 1999

[10] L Brunetti and A M Pastore ldquoLightlike hypersurfaces in inde-ite 119982119982-manifoldsrdquo Differential GeometrymdashDynamical Systemsvol 12 pp 18ndash40 2010

[11] A Bejancu ldquoUmbilical semi-invariant submanifolds of aSasakian manifoldrdquo Tensor vol 37 pp 203ndash213 1982

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Journal of Mathematics 3

and also give

119892119892 10076491007649119860119860119873119873119884119884119884 11988411988411988411988410076651007665 = 119862119862 (119884119884119884 119884119884119884119884) 119884 119892119892 1007649100764911986011986011987311987311988411988411988411987311987310076651007665 = 0119884

119892119892 10076501007650119860119860lowast120585120585119883119883119884 11988411988411988411988410076661007666 = 119861119861 (119883119883119884 119884119884119884119884) 119884 119892119892 10076501007650119860119860lowast

12058512058511988311988311988411987311987310076661007666 = 0(17)

From (9) it is clear that

119861119861 (119883119883119884 120585120585) = 0119884 (18)

that is the second fundamental form of a lightlikehypersurface is degenerate Following the notations ofDuggal and Bejancu [3] for the curvature tensor 119877119877 of119872119872 we have 119877119877(119883119883119884119884119884119884119877119877119884119884119884) = 119892119892(119877119877(119883119883119884 119884119884)119877119877119884119884119884) for any119883119883119884119884119884119884119877119877119884119884119884 119883 119883(119883119883119872119872)

3 Lightlike Hypersurface ofndenite119982119982-Manifolds

Let (119872119872119884 119892119892119884 119872119872(119883119883119872119872)) be a lightlike hypersurface of an indenite119892119892119892119892119892119892-manifold (1198721198722119899119899119899119899119899119884 120601120601119884 120577120577120572120572119884 120578120578

120572120572119884 119892119892) and 120572120572 119883 120572120572119884120572 119884 119899119899120572 suchthat the characteristic vector elds 120577120577120572120572 120572120572 119883 120572120572119884120572 119884 119899119899120572 aretangent to119872119872 As the ambient manifold119872119872 has an additionalgeometric structure 120601120601 we must look for a particular screendistribution on 119872119872 Since 119892119892(120601120601120585120585119884 120585120585) = 0 for any 120585120585 119883 119883(119883119883119872119872⟂)therefore 120601120601(119883119883119872119872⟂) is a distribution on119872119872 of rank 120572 such that119883119883119872119872⟂ cap120601120601(119883119883119872119872⟂) = 1205720120572 Moreover 119892119892(120601120601119873119873119884 120585120585) = 120601119892119892(119873119873119884 120601120601120585120585) = 0for any119873119873 119883 119883(119873119873(119883119883119872119872)) therefore 120601120601119873119873 is tangent to119872119872 Since119892119892(120601120601119873119873119884119873119873) = 0 that is the component of 120601120601119873119873 with respectto 120585120585 vanishes this implies 120601120601119873119873 119883 119883(119872119872(119883119883119872119872)) As 120585120585 and 119873119873 arenull vector elds satisfying 119892119892(120585120585119884119873119873) = 120572 therefore from (2)we deduce that 120601120601120585120585 and 120601120601119873119873 are null vector elds satisfying119892119892(120601120601120585120585119884 120601120601119873119873) = 120572 Hence 120601120601(119883119883119872119872⟂) oplus 120601120601(119873119873(119883119883119872119872)) is a vectorsubbundle of 119872119872(119883119883119872119872) of rank 2 It is known [9] that if structurevector elds 120577120577120572120572 120572120572 119883 120572120572120572 119884 119899119899120572 are tangent to119872119872 then 120577120577120572120572 belongto 119872119872(119883119883119872119872) erefore 119892119892(120601120601120585120585119884 120577120577120572120572) = 119892119892(120601120601119873119873119884 120577120577120572120572) = 0 implies thatthere exists a nondegenerate distribution 1198631198630 of rank 2119899119899 120601 119899on119872119872 such that

119872119872 (119883119883119872119872) = 10076821007682120601120601 (119873119873 (119883119883119872119872)) oplus 120601120601119883119883119872119872⟂10076981007698 ⟂ 1198631198630 ⟂ 1007682100768212057712057712057212057210076981007698 119884 (19)

therefore

119883119883119872119872 = 10076821007682120601120601 (119873119873 (119883119883119872119872)) oplus 120601120601119883119883119872119872⟂10076981007698 ⟂ 1198631198630 ⟂ 1007682100768212057712057712057212057210076981007698 ⟂ 119883119883119872119872⟂(20)

Let 119863119863 = 119883119883119872119872⟂ ⟂ 120601120601119883119883119872119872⟂ ⟂ 1198631198630 and let 119863119863119863 = 120601120601 119873119873(119883119883119872119872) then119863119863 clearly is invariant and119863119863119863 is anti-invariant under 120601120601 and wehave

119883119883119872119872 = 119863119863119863 oplus 119863119863 ⟂ 1007682100768212057712057712057212057210076981007698 (21)

onsider the local lightlike vector elds as

119880119880 = 120601120601120601119873119873 119883 120601120601 (119873119873 (119883119883119872119872)) = 119863119863119863119884

119881119881 = 120601120601120601120585120585 119883 119863119863(22)

Using the decompositions in (21) any 119883119883 119883 119883(119883119883119872119872) can bewritten as

119883119883 = 119872119872119883119883 119899 119883119883119883119883 11989911989911989910055761005576120572120572=120572

120578120578120572120572 (119883119883) 120577120577120572120572119884 (23)

where 119872119872 and 119883119883 are the projection morphisms of 119883119883119872119872 into 119863119863and119863119863119863 respectively andwhere119883119883119883119883 = 119876119876(119883119883)119880119880 and 119876119876 is a local120572-form on119872119872 dened by 119876119876(119883119883) = 119892119892(119881119881119884119883119883) erefore

119876119876 (119880119880) = 120572119884 119876119876 (119883119883) = 0119884 forall119883119883 119883 119883 (119863119863) 119884 1206011206012119873119873 = 120601119873119873 (24)

Applying 120601120601 to (23) and then using (24) we obtain 120601120601119883119883 =120601120601(119872119872119883119883) 119899 119876119876(119883119883)119873119873 By assuming 120601120601(119872119872119883119883) = 120601120601119883119883 for any 119883119883 119883119883(119883119883119872119872) we obtain a tensor eld 120601120601 of type (120572119884 120572) on 119872119872 andgiven by

120601120601119883119883 = 120601120601119883119883 119899 119876119876 (119883119883)119873119873 (25)

Applying 120601120601 to (25) and then using the denition of 120601120601119883119883 weobtain

1206011206012119883119883 = 120601119883119883 119899 119876119876 (119883119883)119880119880 11989911989911989910055761005576120572120572=120572

120578120578120572120572 (119883119883) 120577120577120572120572 (26)

Moreover since 120601120601119883119883 = 120601120601(119872119872119883119883) therefore

120601120601119880119880 = 0119884 120578120578120572120572 ∘ 120601120601 = 0119884 119876119876 1007649100764912060112060111988311988310076651007665 = 0119884 (27)

for any 119883119883 119883 119883(119883119883119872119872) us we have the following theoremanalogous to a theorem proved in [10]

eorem 2 Let (119872119872119884 120601120601119884 120577120577120572120572119884 120578120578120572120572119884 119892119892) be an indenite 119892119892119892119892119892119892-

manifold and let (119872119872119884 119892119892119884 119872119872(119883119883119872119872)) be a lightlike hypersurface of119872119872 then (119872119872119884 120601120601119884 120577120577120572120572119884 119880119880119884 120578120578

120572120572119884 119876119876) is also a 119892119892119892119892119892119892-manifold

Since 120601120601119880119880 = 0 therefore applying 120601120601 to (26) we get 1206011206013 119899120601120601 =0 this implies that 120601120601 is an 119892119892 structure of constant rank on119872119872Now using (2) and (25) we obtain

119892119892 10076491007649120601120601119883119883119884 12060112060111988411988410076651007665 = 119892119892 (119883119883119884 119884119884) 120601 119876119876 (119883119883) 120584120584 (119884119884)

120601 119876119876 (119884119884) 120584120584 (119883119883) 12060111989911989910055761005576120572120572=120572

120598120598120572120572120578120578120572120572 (119883119883) 120578120578120572120572 (119884119884) 119884

(28)

for any119883119883119884119884119884 119883 119883(119883119883119872119872) where 120584120584 is a 120572 form locally dened on119872119872 by 120584120584(120584) = 119892119892(119880119880119884 120584) Replace119883119883 by 120601120601119883119883 in (28) and using (26)and (27) we derive

119892119892 1007649100764912060112060111988311988311988411988411988410076651007665 119899 119892119892 10076491007649119883119883119884 12060112060111988411988410076651007665

= 120601119876119876 (119883119883) 120579120579 (119884119884) 120601 119876119876 (119884119884) 120579120579 (119883119883) (29)

Let 119872119872 be a lightlike hypersurface of an indenite 119982119982-manifold119872119872 then by using (4) (10) and (25) we obtain

nabla119883119883120577120577120572120572 119899 119861119861 10076501007650119883119883119884 12057712057712057212057210076661007666119873119873 = 120601120598120598120572120572120601120601119883119883 120601 120598120598120572120572119876119876 (119883119883)119873119873119884 (30)

4 Journal of Mathematics

for any 119883119883 119883 119883119883119883119883119872119872119872 then comparing the tangential andtransversal components we get

nabla119883119883120577120577120572120572 = minus120598120598120572120572120601120601119883119883120601 (31)

119861119861 10076501007650119883119883120601 12057712057712057212057210076661007666 = minus120598120598120572120572119906119906 119883119883119883119872 120601

this implies ℎ 10076501007650119883119883120601 12057712057712057212057210076661007666 = minus120598120598120572120572119906119906 119883119883119883119872119873119873119873(32)

Using (15) and (31) we get nablalowast119883119883120577120577120572120572 + 119862119862119883119883119883120601 120577120577120572120572119872120585120585 = minus120598120598120572120572120601120601119883119883taking inner product with119873119873 119883 119883119883119873119873119883119883119883119872119872119872119872 we get119862119862119883119883119883120601 120577120577120572120572119872 =minus120598120598120572120572119892119892119883120601120601119883119883120601119873119873119872 and then using (29) we obtain

11986211986210076501007650119883119883120601 12057712057712057212057210076661007666 = 120598120598120572120572119892119892 10076491007649119883119883120601 12060112060111987311987310076651007665 + 119906119906 119883119883119883119872 120579120579 119883119873119873119872 minus 119906119906 119883119873119873119872 120579120579 119883119883119883119872

= minus120598120598120572120572119892119892 119883119883119883120601119883119883119872

= minus120598120598120572120572120584120584 119883119883119883119872 119873

(33)

Using (3) for any119873119873 119883 119883119883119873119873119883119883119883119872119872119872119872 we get nabla119883119883120601120601119873119873 = 120601120601nabla119883119883119873119873 +119892119892119883119883119883120601119873119873119872120577120577 therefore

119861119861 119883119883119883120601119883119883119872 = 119892119892 119883ℎ 119883119883119883120601119883119883119872 120601 120585120585119872

= minus119892119892 10076501007650nabla119883119883120601120601119873119873120601 12058512058510076661007666

= 119892119892 10076501007650nabla119883119883120601 12060112060112058512058510076661007666

= 119892119892 1007649100764911986011986011987311987311988311988312060111988311988310076651007665

= 119862119862 119883119883119883120601119883119883119872 119873

(34)

4 Totally Contact Umbilical Lightlikeyesface of ndenite119982119982-Manifolds

ere wewill follow eancus denition [11] of totally contactumbilical submanifolds of Sasakian manifolds to state totallycontact umbilical lightlike hypersurfaces of indenite 119982119982-manifolds

e A lightlike hypersurface 119872119872 of an indenite119982119982-manifold is said to be totally contact umbilical lightlikehypersurface if the second fundamental form ℎ of119872119872 satises

ℎ 119883119883119883120601 119883119883119872 = 10076871007687119892119892 119883119883119883120601119883119883119872 minus11990311990310055761005576120572120572=120572

120598120598120572120572120578120578120572120572 119883119883119883119872 120578120578120572120572 11988311988311988311987210077031007703119867119867

+11990311990310055761005576120572120572=120572

120578120578120572120572 119883119883119883119872 ℎ 10076501007650119883119883120601 12057712057712057212057210076661007666

+11990311990310055761005576120572120572=120572

120578120578120572120572 119883119883119883119872 ℎ 10076501007650119883119883120601 12057712057712057212057210076661007666 120601

(35)

for any 119883119883120601119883119883 119883 119883119883119883119883119872119872119872 where119867119867 is a transversal vector eldon 119872119872 that is 119867119867 = 119867119867119873119873 where 119867119867 is a smooth function on119984119984 119984 119872119872

Remark 4 We can also write (35) asℎ 119883119883119883120601 119883119883119872 = 119861119861 119883119883119883120601 119883119883119872119873119873

= 10076811007681119861119861120572 119883119883119883120601 119883119883119872 + 1198611198612 119883119883119883120601 11988311988311987210076971007697119873119873120601(36)

where

119861119861120572 119883119883119883120601 119883119883119872 = 11986711986710076871007687119892119892 119883119883119883120601119883119883119872 minus11990311990310055761005576120572120572=120572

120598120598120572120572120578120578120572120572 119883119883119883119872 120578120578120572120572 11988311988311988311987210077031007703 120601 (37)

and using (32)

1198611198612 119883119883119883120601 119883119883119872 = minus120578120578 119883119883119883119872 119906119906 119883119883119883119872 minus 120578120578 119883119883119883119872 119906119906 119883119883119883119872 120601 (38)

for any119883119883120601119883119883 119883 119883119883119883119883119872119872119872 If 119867119867 = 120582 that is 119861119861120572 = 120582 then lightlikehypersurface119872119872 is said to be totally contact geodesic

Let 119877119877 be the curvature tensor elds of 119872119872119872 then for anindenite 119982119982-space form119872119872119883119872119872119872 we have (see [8])

119877119877 119883119883119883120601119883119883120601119883119883120601119883119883119872

=119872119872 + 1198881205981205984

10076821007682119892119892 10076501007650120601120601119883119883120601 12060112060111988311988310076661007666 119892119892 10076501007650120601120601119883119883120601 12060112060111988311988310076661007666

minus119892119892 10076501007650120601120601119883119883120601 12060112060111988311988310076661007666 119892119892 10076501007650120601120601119883119883120601 1206011206011198831198831007666100766610076981007698

+119872119872 minus 1205981205984

Φ 119883119883119883120601119883119883119872Φ 119883119883119883120601 119883119883119872 minus Φ 119883119883119883120601119883119883119872Φ 119883119883119883120601119883119883119872

+2Φ 119883119883119883120601119883119883119872Φ 119883119883119883120601119883119883119872

+ 10076821007682120578120578 119883119883119883119872 120578120578 119883119883119883119872 119892119892 10076501007650120601120601119883119883120601 12060112060111988311988310076661007666 minus 120578120578 119883119883119883119872 120578120578 119883119883119883119872 119892119892 10076501007650120601120601119883119883120601 12060112060111988311988310076661007666

+ 120578120578 119883119883119883119872 120578120578 119883119883119883119872 119892119892 10076501007650120601120601119883119883120601 12060112060111988311988310076661007666

minus120578120578 119883119883119883119872 120578120578 119883119883119883119872 119892119892 10076501007650120601120601119883119883120601 1206011206011198831198831007666100766610076981007698 120601(39)

for any119883119883120601119883119883120601119883119883120601119883119883 119883 119883119883119883119883119872119872119872 whereΦ119883119883119883120601119883119883119872 = 119892119892119883119883119883120601 120601120601119883119883119872 and120598120598 = 120598119903119903

120572120572=120572 120598120598120572120572 Using (39) we obtain

119877119877 119883119883119883120601119883119883120601119883119883120601 120585120585119872 =119872119872 minus 1205981205984

10076821007682119892119892 119883119883119883120601119883119883119872 119892119892 10076501007650119883119883120601 12060112060111988311988310076661007666

minus 119892119892 119883119883119883120601 119883119883119872 119892119892 10076501007650119883119883120601 12060112060111988311988310076661007666

+2119892119892 119883119883119883120601119883119883119872 119892119892 10076501007650119883119883120601 1206011206011198831198831007666100766610076981007698 119873

(40)

Also for the pair 120585120585120601119873119873 on119984119984 119984 119872119872 from (38) of page no94 of [3] we have

119877119877 119883119883119883120601119883119883120601119883119883120601 120585120585119872 = 10076491007649nabla11988311988311986111986110076651007665 119883119883119883120601119883119883119872 minus 10076491007649nabla11988311988311986111986110076651007665 119883119883119883120601119883119883119872

+ 120591120591 119883119883119883119872 119861119861 119883119883119883120601119883119883119872 minus 120591120591 119883119883119883119872 119861119861 119883119883119883120601119883119883119872 120601(41)

119883119883120601119883119883120601119883119883 119883 119883119883119883119883119872119872119872 and 120585120585 119883 119883119883119883119883119872119872⟂119872 where

10076491007649nabla11988311988311986111986110076651007665 119883119883119883120601119883119883119872 = 119883119883 119883 119861119861 119883119883119883120601119883119883119872 minus 119861119861 10076491007649nabla11988311988311988311988312060111988311988310076651007665

minus 119861119861 10076491007649119883119883120601 nabla11988311988311988311988310076651007665 119873(42)

Journal of Mathematics 5

Hence using (40) and (41) we obtain

10076491007649nabla11988311988311986111986110076651007665 1007665119884119884119884119884119884) minus 10076491007649nabla11988411988411986111986110076651007665 1007665119883119883119884119884119884)

=119888119888 minus 1198881198884

10076821007682119892119892 1007665119881119881119884119883119883) 119892119892 10076501007650119884119884119884 12060112060111988411988410076661007666 minus 119892119892 1007665119881119881119884 119884119884) 119892119892 10076501007650119884119884119884 12060112060111988311988310076661007666

+2119892119892 1007665119881119881119884119884119884) 119892119892 10076501007650119883119883119884 1206011206011198841198841007666100766610076661007666

minus 120591120591 1007665119883119883) 119861119861 1007665119884119884119884119884119884) + 120591120591 1007665119884119884) 119861119861 1007665119883119883119884119884119884) 119884

(43)

for any119883119883119884119884119884119884119884119884 119883 1198831007665119883119883119872119872)

Lemma 5 Let 1007665119872119872119884 119892119892) be a totally contact umbilical lightlikehypersurface of an indenite 119982119982-manifold 1007665119872119872119884 119892119892) then for any119883119883119884119884119884119884119884119884 119883 1198831007665119883119883119872119872) one has

10076491007649nabla119883119883119861119861110076651007665 1007665119884119884119884119884119884) = 1007665119883119883 119883 119883119883)10076871007687119892119892 1007665119884119884119884119884119884) minus11990311990310055761005576120572120572=1

119888119888120572120572120578120578120572120572 1007665119884119884) 120578120578120572120572 1007665119884119884)10077031007703

+ 119883119883 119861119861 1007665119883119883119884 119884119884) 120579120579 1007665119884119884) + 119861119861 1007665119883119883119884119884119884) 120579120579 1007665119884119884)

+ 119883119883120578120578 1007665119884119884) 10076811007681119892119892 1007649100764912060112060111988311988311988411988411988410076651007665 + 119906119906 1007665119883119883) 120579120579 1007665119884119884)10076971007697

+ 119883119883120578120578 1007665119884119884) 10076811007681119892119892 1007649100764912060112060111988311988311988411988411988410076651007665 + 119906119906 1007665119883119883) 120579120579 1007665119884119884)10076971007697

10076491007649nabla119883119883119861119861210076651007665 1007665119884119884119884119884119884) = 119888119888119906119906 1007665119884119884) 10076811007681119892119892 10076491007649119884119884119884 12060112060111988311988310076651007665 + 119906119906 1007665119883119883) 120579120579 1007665119884119884)10076971007697

+ 119888119888119906119906 1007665119884119884) 10076811007681119892119892 10076491007649119884119884119884 12060112060111988311988310076651007665 + 119906119906 1007665119883119883) 120579120579 1007665119884119884)10076971007697

+ 120578120578 1007665119884119884) 10076811007681119861119861 10076491007649119883119883119884 12060112060111988411988410076651007665 + 120591120591 1007665119883119883) 119906119906 1007665119884119884)10076971007697

+ 120578120578 1007665119884119884) 10076811007681119861119861 10076491007649119883119883119884 12060112060111988411988410076651007665 + 120591120591 1007665119883119883) 119906119906 1007665119884119884)10076971007697 (44)

Proof By virtue of (37) and (42) we obtain

10076491007649nabla119883119883119861119861110076651007665 1007665119884119884119884119884119884) = 1007665119883119883 119883 119883119883)10076871007687119892119892 1007665119884119884119884119884119884) minus11990311990310055761005576120572120572=1

119888119888120572120572120578120578120572120572 1007665119884119884) 120578120578120572120572 1007665119884119884)10077031007703

minus 119883119883 10076811007681119892119892 10076491007649nabla11988311988311988411988411988411988411988410076651007665 + 119892119892 10076491007649119884119884119884 nabla1198831198831198841198841007665100766510076971007697

+ 11988311988311990311990310055761005576120572120572=1

119888119888120572120572 10076811007681120578120578120572120572 10076491007649nabla11988311988311988411988410076651007665 120578120578

120572120572 1007665119884119884)

+120578120578120572120572 10076491007649nabla11988311988311988411988410076651007665 120578120578120572120572 1007665119884119884)10076971007697 (45)

Now using (12) in (45) we have

10076491007649nabla119883119883119861119861110076651007665 1007665119884119884119884119884119884) = 1007665119883119883 119883 119883119883)10076871007687119892119892 1007665119884119884119884119884119884) minus11990311990310055761005576120572120572=1

119888119888120572120572120578120578120572120572 1007665119884119884) 120578120578120572120572 1007665119884119884)10077031007703

minus 119883119883 10076811007681119883119883 10076491007649119892119892 1007665119884119884119884119884119884)10076651007665 minus 119861119861 1007665119883119883119884 119884119884) 120579120579 1007665119884119884)

minus119861119861 1007665119883119883119884119884119884) 120579120579 1007665119884119884)

+ 11988311988311990311990310055761005576120572120572=1

10076821007682119892119892 10076501007650nabla119883119883119884119884119884 12057712057712057212057210076661007666 120578120578120572120572 1007665119884119884)

+119892119892 10076501007650nabla119883119883119884119884119884 12057712057712057212057210076661007666 120578120578120572120572 1007665119884119884)10076661007666

(46)

Again using (12) we have

119892119892 10076501007650nabla119883119883119884119884119884 12057712057712057212057210076661007666 = 119883119883 10076501007650119892119892 10076501007650119884119884119884 1205771205771205721205721007666100766610076661007666 + 119888119888120572120572119892119892 10076491007649119884119884119884 12060112060111988311988310076651007665

+ 119888119888120572120572119906119906 1007665119883119883) 120579120579 1007665119884119884) (47)

us from (46) and (47) the rst expression of the theoremfollows Next using (38) and (42) we obtain

10076491007649nabla119883119883119861119861210076651007665 1007665119884119884119884119884119884) =11990311990310055761005576120572120572=1

119888119888120572120572 10076811007681120578120578120572120572 10076491007649nabla11988311988311988411988410076651007665 119906119906 1007665119884119884) + 120578120578120572120572 10076491007649nabla11988311988311988411988410076651007665119906119906 1007665119884119884)

+ 120578120578120572120572 1007665119884119884) 119906119906 10076491007649nabla11988311988311988411988410076651007665

+120578120578120572120572 1007665119884119884) 119906119906 10076491007649nabla1198831198831198841198841007665100766510076971007697 (48)

Using (31) and (32) we get

120578120578120572120572 10076491007649nabla11988311988311988411988410076651007665 = 119892119892 10076491007649119884119884119884 12060112060111988311988310076651007665 + 119906119906 1007665119883119883) 120579120579 1007665119884119884) 119884 (49)

and from (2) (3) (10) (16) (17) (18) and (25) we obtain

119906119906 10076491007649nabla11988311988311988411988410076651007665 = 119861119861 10076491007649119883119883119884 12060112060111988411988410076651007665 + 120591120591 1007665119883119883) 119906119906 1007665119884119884) (50)

us by substituting (49) and (50) in (48) we obtain (44)

eorem 6 Let 1198721198721007665119888119888) be an indenite 119982119982-space form and 119872119872be a totally contact umbilical lightlike hypersurface of 1198721198721007665119888119888)en 119888119888 = minus119888119888119888 where 119888119888 = 120598119903119903

120572120572=1 119888119888120572120572 and 119883119883 satises the followingdifferential equations

120585120585 119883 119883119883 + 119883119883120591120591 1007665120585120585) minus 1198831198832 = 0119884

119875119875119883119883 119883 119883119883 + 119883119883120591120591 1007665119875119875119883119883) = 0119884(51)

for any119883119883 119883 1198831007665119883119883119872119872)

Proof Let 119872119872 be a totally contact umbilical lightlike hyper-surface of an indenite 119982119982-space form 1198721198721007665119888119888) of constant 120601120601-sectional curvature 119888119888 en using (35) we have 1198611198611007665119883119883119884 120601120601119884119884) =1198831198831198921198921007665119883119883119884 120601120601119884119884) Substituting (29) (44) in (43) we obtain

1007665119883119883 119883 119883119883)10076871007687119892119892 1007665119884119884119884119884119884) minus11990311990310055761005576120572120572=1

119888119888120572120572120578120578120572120572 1007665119884119884) 120578120578120572120572 1007665119884119884)10077031007703

minus 1007665119884119884 119883 119883119883)10076871007687119892119892 1007665119883119883119884119884119884) minus11990311990310055761005576120572120572=1

119888119888120572120572120578120578120572120572 1007665119883119883) 120578120578120572120572 1007665119884119884)10077031007703

+ 119883119883 119861119861 1007665119883119883119884119884119884) 120579120579 1007665119884119884) minus 119861119861 1007665119884119884119884119884119884) 120579120579 1007665119883119883)

+ 2119883119883 10076811007681119892119892 1007649100764912060112060111988311988311988411988411988410076651007665 + 119906119906 1007665119883119883) 120579120579 1007665119884119884)10076971007697 120578120578 1007665119884119884)

+ 119883119883 1007681100768110076491007649119892119892 10076491007649119884119884119884 12060112060111988311988310076651007665 120578120578 1007665119884119884) minus 119892119892 10076491007649119884119884119884 12060112060111988411988410076651007665 120578120578 1007665119883119883)10076971007697

+ 119883119883 10076811007681119906119906 1007665119883119883) 120578120578 1007665119884119884) minus 119906119906 1007665119884119884) 120578120578 1007665119883119883)10076971007697 120579120579 1007665119884119884)

+ 2119888119888 10076811007681119892119892 1007649100764912060112060111988311988311988411988411988410076651007665 + 119906119906 1007665119883119883) 120579120579 1007665119884119884)10076971007697 119906119906 1007665119884119884)

+ 10076811007681119883119883119892119892 10076491007649119883119883119884 12060112060111988411988410076651007665 + 120591120591 1007665119883119883) 119906119906 1007665119884119884)

minus119883119883119892119892 10076491007649119884119884119884 12060112060111988311988310076651007665 minus 120591120591 1007665119884119884) 119906119906 1007665119883119883)10076971007697 120578120578 1007665119884119884)

6 Journal of Mathematics

+ 120582120582 10076811007681119892119892 10076491007649119883119883119883 11988311988311988311988310076651007665 120578120578 (119884119884) minus 119892119892 10076491007649119884119884119883 11988311988311988311988310076651007665 120578120578 (119883119883)10076971007697

+ 10076811007681120591120591 (119883119883) 120578120578 (119884119884) minus 120591120591 (119884119884) 120578120578 (119883119883)10076971007697 119906119906 (119883119883)

+ 120598120598 10076811007681119892119892 10076491007649119883119883119883 11988311988311988311988310076651007665 119906119906 (119884119884) minus 119892119892 10076491007649119883119883119883 11988311988311988411988410076651007665 119906119906 (119883119883)10076971007697

=119888119888 minus 1205981205984

10076821007682119892119892 (119881119881119883119883119883) 119892119892 10076501007650119883119883119883 11988311988311988411988410076661007666

minus 119892119892 (119881119881119883 119884119884) 119892119892 10076501007650119883119883119883 11988311988311988311988310076661007666

+2119892119892 (119881119881119883119883119883) 119892119892 10076501007650119883119883119883 1198831198831198841198841007666100766610076981007698

+ 120591120591 (119884119884) 119861119861 (119883119883119883119883119883) minus 120591120591 (119883119883) 119861119861 (119884119884119883119883119883) 119883(52)

for any119883119883119883119884119884119883119883119883 119883 119883(119883119883119872119872) Put119883119883 = 119883119883 in (52) we get

(119883119883 120585 120582120582) 10076871007687119892119892 (119884119884119883119883119883) minus11990311990310055761005576120572120572=120572

120598120598120572120572120578120578120572120572 (119884119884) 120578120578120572120572 (119883119883)10077031007703

minus 120582120582119861119861 (119884119884119883119883119883) minus 2120582120582119906119906 (119884119884) 120578120578 (119883119883)

minus 120582120582119906119906 (119883119883) 120578120578 (119884119884) minus 2120598120598119906119906 (119884119884) 119906119906 (119883119883)

+ 120591120591 (119883119883) 119906119906 (119884119884) 120578120578 (119883119883) + 2120582120582119906119906 (119884119884) 120578120578 (119883119883)

+ 120591120591 (119883119883) 119906119906 (119883119883) 120578120578 (119884119884) minus 120598120598119906119906 (119883119883) 119906119906 (119884119884)

+ 120582120582119906119906 (119883119883) 120578120578 (119884119884)

=119888119888 minus 1205981205984

119906119906 (119884119884) 119906119906 (119883119883) + 2119906119906 (119883119883) 119906119906 (119884119884)

minus 120591120591 (119883119883) 119861119861 (119884119884119883119883119883)

(53)

Put 119884119884 = 119883119883 = 119884119884 in (53) and then using 119892119892(119884119884119883119884119884) = 119892 119906119906(119884119884) =120572 120578120578120572120572(119884119884) = 119892 we obtain

minus120582120582119861119861 (119884119884119883119884119884) minus 3120598120598 = 3(119888119888 minus 120598120598)4

minus 120591120591 (119883119883) 119861119861 (119884119884119883119884119884) 119883 (54)

since 119861119861(119884119884119883119884119884) = 119892 we get 119888119888 = minus3120598120598 Moreover by putting 119884119884 =119881119881 and119883119883 = 119884119884 in (52) we obtain

119883119883 120585 120582120582 + 120582120582120591120591 (119883119883) minus 1205821205822 = 119892 (55)

Finally putting119883119883 = 119883119883119883119883 119884119884 = 119883119883119884119884 and119883119883 = 119883119883119883119883 in (52) with119888119888 = minus3120598120598 and using that 119878119878(119883119883119872119872) is nondegenerate we obtain

119883119883119883119883 120585 120582120582 + 120582120582120591120591 (119883119883119883119883)10076551007655119883119883119884119884 minus11990311990310055761005576120572120572=120572

120578120578120572120572 (119883119883119884119884) 12057712057712057212057210076711007671

= 119883119883119884119884 120585 120582120582 + 120582120582120591120591 (119883119883119884119884)10076551007655119883119883119883119883 minus11990311990310055761005576120572120572=120572

120578120578120572120572 (119883119883119883119883) 12057712057712057212057210076711007671

(56)

Putting 119883119883119883119883 = 120577120577120572120572 in (56) we get

10076821007682120577120577120572120572 120585 120582120582 + 120582120582120591120591 10076501007650120577120577120572120572100766610076661007698100769810076551007655119883119883119884119884 minus11990311990310055761005576120572120572=120572

120578120578120572120572 (119883119883119884119884) 12057712057712057212057210076711007671 = 119892 (57)

By taking 119884119884 = 119881119881 we obtain

120577120577120572120572 120585 120582120582 + 120582120582120591120591 1007650100765012057712057712057212057210076661007666 = 119892 (58)

Since 119883119883119883119883 119883 119883(119878119878(119883119883119872119872)) therefore using (19) we can write

119883119883119883119883 = 119906119906 (119883119883119883119883)119884119884 + 119880119880 (119883119883119883119883)119881119881 +2119899119899minus410055761005576119894119894=120572

120573120573119894119894119865119865119894119894

+11990311990310055761005576120572120572=120572

120598120598120572120572120578120578120572120572 (119883119883119883119883) 120577120577120572120572

= 119883119883119883119883prime +11990311990310055761005576120572120572=120572

120598120598120572120572120578120578120572120572 (119883119883119883119883) 120577120577120572120572119883

(59)

where 119865119865119894119894120572le119894119894le2119899119899minus4 is an orthogonal basis of 119863119863119892 then using(58) we have

119883119883119883119883 120585 120582120582 + 120582120582120591120591 (119883119883119883119883) = 10076551007655119883119883119883119883prime +11990311990310055761005576120572120572=120572

120598120598120572120572120578120578120572120572 (119883119883119883119883) 12057712057712057212057210076711007671 120585 120582120582

+ 12058212058212059112059110076551007655119883119883119883119883prime +11990311990310055761005576120572120572=120572

120598120598120572120572120578120578120572120572 (119883119883119883119883) 12057712057712057212057210076711007671

= 119883119883119883119883prime 120585 120582120582 + 120582120582120591120591 10076511007651119883119883119883119883prime10076671007667 119883

(60)

which leads to get from (56)

10076831007683119883119883119883119883prime 120585 120582120582 + 120582120582120591120591 10076511007651119883119883119883119883prime1007667100766710076991007699119883119883119884119884prime = 10076831007683119883119883119884119884prime 120585 120582120582 + 120582120582120591120591 10076511007651119883119883119884119884prime1007667100766710076991007699119883119883119883119883prime(61)

Now assume that there eists a vector eld 119883119883119892 on someneighborhood of119872119872 such that 119883119883119883119883prime

119892 120585 120582120582 + 120582120582120591120591(119883119883119883119883prime119892) ne 119892 at some

point 119901119901 in the neighborhood en from (61) it is clear thatall the vectors of the ber (119878119878(119883119883119872119872) minus 120577120577120572120572)119901119901 are collinear with(119883119883119883119883prime

119892)119901119901is contradicts dim(119878119878(119883119883119872119872)minus120577120577120572120572) gt 120572is impliesthe result

From the previous mentioned theorem we have thefollowing corollary

Corollary 7 ere exist no totally contact umbilical lightlikereal hypersurfaces of indenite 119982119982-space form119872119872(119888119888) (119888119888 ne minus 3120598120598)with 120577120577120572120572 119883 119883119883119872119872

Acknowledgment

e authors would like to thank the anonymous referee forhisher valuable suggestions that helped them to improve thispaper

References

[1] B Y Chen Geometry of Submanifolds Marcel Dekker NewYork USA 1973

[2] B OrsquoNeill Semi-Riemannian Geometry with Applications toRelativity Academic Press 1983

[3] K L Duggal and A Bejancu Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications Kluwer AcademicPublishers Dordrecht e Netherlands 1996

Journal of Mathematics 7

[4] L Duggal and B Sahin ldquoLightlike submanifolds of indeniteSasakian manifoldsrdquo International Journal of Mathematics andMathematical Sciences vol 2007 Article ID 57585 21 pages2007

[5] V I Arnold ldquoContact geometry the geometrical method ofGibbss thermodynamicsrdquo in Proceedings of the Gibbs Sym-posium pp 163ndash179 American Mathematical Society NewHaven Conn USA 1989

[6] SMaclaneGeometricalMechanics II Lecture Notes Universityof Chicago Chicago Ill USA 1968

[7] V E Nazaikinskii V E Shatalov and B Y Sternin ContactGeometry and Linear Differential Equations vol 6 of DeGruyter Expositions in Mathematics Walter de Gruyter BerlinGermany 1992

[8] L Brunetti and AM Pastore ldquoCurvature of a class of indeniteglobally framed f-manifoldsrdquo Bulletin Mathematique de laSociete des Sciences Mathematiques de Roumanie vol 51 no 3pp 183ndash204 2008

[9] C Calin ldquoOn existence of degenerate hypersurfaces in Sasakianmanifoldsrdquo Arab Journal of Mathematical Sciences vol 5 pp21ndash27 1999

[10] L Brunetti and A M Pastore ldquoLightlike hypersurfaces in inde-ite 119982119982-manifoldsrdquo Differential GeometrymdashDynamical Systemsvol 12 pp 18ndash40 2010

[11] A Bejancu ldquoUmbilical semi-invariant submanifolds of aSasakian manifoldrdquo Tensor vol 37 pp 203ndash213 1982

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4 Journal of Mathematics

for any 119883119883 119883 119883119883119883119883119872119872119872 then comparing the tangential andtransversal components we get

nabla119883119883120577120577120572120572 = minus120598120598120572120572120601120601119883119883120601 (31)

119861119861 10076501007650119883119883120601 12057712057712057212057210076661007666 = minus120598120598120572120572119906119906 119883119883119883119872 120601

this implies ℎ 10076501007650119883119883120601 12057712057712057212057210076661007666 = minus120598120598120572120572119906119906 119883119883119883119872119873119873119873(32)

Using (15) and (31) we get nablalowast119883119883120577120577120572120572 + 119862119862119883119883119883120601 120577120577120572120572119872120585120585 = minus120598120598120572120572120601120601119883119883taking inner product with119873119873 119883 119883119883119873119873119883119883119883119872119872119872119872 we get119862119862119883119883119883120601 120577120577120572120572119872 =minus120598120598120572120572119892119892119883120601120601119883119883120601119873119873119872 and then using (29) we obtain

11986211986210076501007650119883119883120601 12057712057712057212057210076661007666 = 120598120598120572120572119892119892 10076491007649119883119883120601 12060112060111987311987310076651007665 + 119906119906 119883119883119883119872 120579120579 119883119873119873119872 minus 119906119906 119883119873119873119872 120579120579 119883119883119883119872

= minus120598120598120572120572119892119892 119883119883119883120601119883119883119872

= minus120598120598120572120572120584120584 119883119883119883119872 119873

(33)

Using (3) for any119873119873 119883 119883119883119873119873119883119883119883119872119872119872119872 we get nabla119883119883120601120601119873119873 = 120601120601nabla119883119883119873119873 +119892119892119883119883119883120601119873119873119872120577120577 therefore

119861119861 119883119883119883120601119883119883119872 = 119892119892 119883ℎ 119883119883119883120601119883119883119872 120601 120585120585119872

= minus119892119892 10076501007650nabla119883119883120601120601119873119873120601 12058512058510076661007666

= 119892119892 10076501007650nabla119883119883120601 12060112060112058512058510076661007666

= 119892119892 1007649100764911986011986011987311987311988311988312060111988311988310076651007665

= 119862119862 119883119883119883120601119883119883119872 119873

(34)

4 Totally Contact Umbilical Lightlikeyesface of ndenite119982119982-Manifolds

ere wewill follow eancus denition [11] of totally contactumbilical submanifolds of Sasakian manifolds to state totallycontact umbilical lightlike hypersurfaces of indenite 119982119982-manifolds

e A lightlike hypersurface 119872119872 of an indenite119982119982-manifold is said to be totally contact umbilical lightlikehypersurface if the second fundamental form ℎ of119872119872 satises

ℎ 119883119883119883120601 119883119883119872 = 10076871007687119892119892 119883119883119883120601119883119883119872 minus11990311990310055761005576120572120572=120572

120598120598120572120572120578120578120572120572 119883119883119883119872 120578120578120572120572 11988311988311988311987210077031007703119867119867

+11990311990310055761005576120572120572=120572

120578120578120572120572 119883119883119883119872 ℎ 10076501007650119883119883120601 12057712057712057212057210076661007666

+11990311990310055761005576120572120572=120572

120578120578120572120572 119883119883119883119872 ℎ 10076501007650119883119883120601 12057712057712057212057210076661007666 120601

(35)

for any 119883119883120601119883119883 119883 119883119883119883119883119872119872119872 where119867119867 is a transversal vector eldon 119872119872 that is 119867119867 = 119867119867119873119873 where 119867119867 is a smooth function on119984119984 119984 119872119872

Remark 4 We can also write (35) asℎ 119883119883119883120601 119883119883119872 = 119861119861 119883119883119883120601 119883119883119872119873119873

= 10076811007681119861119861120572 119883119883119883120601 119883119883119872 + 1198611198612 119883119883119883120601 11988311988311987210076971007697119873119873120601(36)

where

119861119861120572 119883119883119883120601 119883119883119872 = 11986711986710076871007687119892119892 119883119883119883120601119883119883119872 minus11990311990310055761005576120572120572=120572

120598120598120572120572120578120578120572120572 119883119883119883119872 120578120578120572120572 11988311988311988311987210077031007703 120601 (37)

and using (32)

1198611198612 119883119883119883120601 119883119883119872 = minus120578120578 119883119883119883119872 119906119906 119883119883119883119872 minus 120578120578 119883119883119883119872 119906119906 119883119883119883119872 120601 (38)

for any119883119883120601119883119883 119883 119883119883119883119883119872119872119872 If 119867119867 = 120582 that is 119861119861120572 = 120582 then lightlikehypersurface119872119872 is said to be totally contact geodesic

Let 119877119877 be the curvature tensor elds of 119872119872119872 then for anindenite 119982119982-space form119872119872119883119872119872119872 we have (see [8])

119877119877 119883119883119883120601119883119883120601119883119883120601119883119883119872

=119872119872 + 1198881205981205984

10076821007682119892119892 10076501007650120601120601119883119883120601 12060112060111988311988310076661007666 119892119892 10076501007650120601120601119883119883120601 12060112060111988311988310076661007666

minus119892119892 10076501007650120601120601119883119883120601 12060112060111988311988310076661007666 119892119892 10076501007650120601120601119883119883120601 1206011206011198831198831007666100766610076981007698

+119872119872 minus 1205981205984

Φ 119883119883119883120601119883119883119872Φ 119883119883119883120601 119883119883119872 minus Φ 119883119883119883120601119883119883119872Φ 119883119883119883120601119883119883119872

+2Φ 119883119883119883120601119883119883119872Φ 119883119883119883120601119883119883119872

+ 10076821007682120578120578 119883119883119883119872 120578120578 119883119883119883119872 119892119892 10076501007650120601120601119883119883120601 12060112060111988311988310076661007666 minus 120578120578 119883119883119883119872 120578120578 119883119883119883119872 119892119892 10076501007650120601120601119883119883120601 12060112060111988311988310076661007666

+ 120578120578 119883119883119883119872 120578120578 119883119883119883119872 119892119892 10076501007650120601120601119883119883120601 12060112060111988311988310076661007666

minus120578120578 119883119883119883119872 120578120578 119883119883119883119872 119892119892 10076501007650120601120601119883119883120601 1206011206011198831198831007666100766610076981007698 120601(39)

for any119883119883120601119883119883120601119883119883120601119883119883 119883 119883119883119883119883119872119872119872 whereΦ119883119883119883120601119883119883119872 = 119892119892119883119883119883120601 120601120601119883119883119872 and120598120598 = 120598119903119903

120572120572=120572 120598120598120572120572 Using (39) we obtain

119877119877 119883119883119883120601119883119883120601119883119883120601 120585120585119872 =119872119872 minus 1205981205984

10076821007682119892119892 119883119883119883120601119883119883119872 119892119892 10076501007650119883119883120601 12060112060111988311988310076661007666

minus 119892119892 119883119883119883120601 119883119883119872 119892119892 10076501007650119883119883120601 12060112060111988311988310076661007666

+2119892119892 119883119883119883120601119883119883119872 119892119892 10076501007650119883119883120601 1206011206011198831198831007666100766610076981007698 119873

(40)

Also for the pair 120585120585120601119873119873 on119984119984 119984 119872119872 from (38) of page no94 of [3] we have

119877119877 119883119883119883120601119883119883120601119883119883120601 120585120585119872 = 10076491007649nabla11988311988311986111986110076651007665 119883119883119883120601119883119883119872 minus 10076491007649nabla11988311988311986111986110076651007665 119883119883119883120601119883119883119872

+ 120591120591 119883119883119883119872 119861119861 119883119883119883120601119883119883119872 minus 120591120591 119883119883119883119872 119861119861 119883119883119883120601119883119883119872 120601(41)

119883119883120601119883119883120601119883119883 119883 119883119883119883119883119872119872119872 and 120585120585 119883 119883119883119883119883119872119872⟂119872 where

10076491007649nabla11988311988311986111986110076651007665 119883119883119883120601119883119883119872 = 119883119883 119883 119861119861 119883119883119883120601119883119883119872 minus 119861119861 10076491007649nabla11988311988311988311988312060111988311988310076651007665

minus 119861119861 10076491007649119883119883120601 nabla11988311988311988311988310076651007665 119873(42)

Journal of Mathematics 5

Hence using (40) and (41) we obtain

10076491007649nabla11988311988311986111986110076651007665 1007665119884119884119884119884119884) minus 10076491007649nabla11988411988411986111986110076651007665 1007665119883119883119884119884119884)

=119888119888 minus 1198881198884

10076821007682119892119892 1007665119881119881119884119883119883) 119892119892 10076501007650119884119884119884 12060112060111988411988410076661007666 minus 119892119892 1007665119881119881119884 119884119884) 119892119892 10076501007650119884119884119884 12060112060111988311988310076661007666

+2119892119892 1007665119881119881119884119884119884) 119892119892 10076501007650119883119883119884 1206011206011198841198841007666100766610076661007666

minus 120591120591 1007665119883119883) 119861119861 1007665119884119884119884119884119884) + 120591120591 1007665119884119884) 119861119861 1007665119883119883119884119884119884) 119884

(43)

for any119883119883119884119884119884119884119884119884 119883 1198831007665119883119883119872119872)

Lemma 5 Let 1007665119872119872119884 119892119892) be a totally contact umbilical lightlikehypersurface of an indenite 119982119982-manifold 1007665119872119872119884 119892119892) then for any119883119883119884119884119884119884119884119884 119883 1198831007665119883119883119872119872) one has

10076491007649nabla119883119883119861119861110076651007665 1007665119884119884119884119884119884) = 1007665119883119883 119883 119883119883)10076871007687119892119892 1007665119884119884119884119884119884) minus11990311990310055761005576120572120572=1

119888119888120572120572120578120578120572120572 1007665119884119884) 120578120578120572120572 1007665119884119884)10077031007703

+ 119883119883 119861119861 1007665119883119883119884 119884119884) 120579120579 1007665119884119884) + 119861119861 1007665119883119883119884119884119884) 120579120579 1007665119884119884)

+ 119883119883120578120578 1007665119884119884) 10076811007681119892119892 1007649100764912060112060111988311988311988411988411988410076651007665 + 119906119906 1007665119883119883) 120579120579 1007665119884119884)10076971007697

+ 119883119883120578120578 1007665119884119884) 10076811007681119892119892 1007649100764912060112060111988311988311988411988411988410076651007665 + 119906119906 1007665119883119883) 120579120579 1007665119884119884)10076971007697

10076491007649nabla119883119883119861119861210076651007665 1007665119884119884119884119884119884) = 119888119888119906119906 1007665119884119884) 10076811007681119892119892 10076491007649119884119884119884 12060112060111988311988310076651007665 + 119906119906 1007665119883119883) 120579120579 1007665119884119884)10076971007697

+ 119888119888119906119906 1007665119884119884) 10076811007681119892119892 10076491007649119884119884119884 12060112060111988311988310076651007665 + 119906119906 1007665119883119883) 120579120579 1007665119884119884)10076971007697

+ 120578120578 1007665119884119884) 10076811007681119861119861 10076491007649119883119883119884 12060112060111988411988410076651007665 + 120591120591 1007665119883119883) 119906119906 1007665119884119884)10076971007697

+ 120578120578 1007665119884119884) 10076811007681119861119861 10076491007649119883119883119884 12060112060111988411988410076651007665 + 120591120591 1007665119883119883) 119906119906 1007665119884119884)10076971007697 (44)

Proof By virtue of (37) and (42) we obtain

10076491007649nabla119883119883119861119861110076651007665 1007665119884119884119884119884119884) = 1007665119883119883 119883 119883119883)10076871007687119892119892 1007665119884119884119884119884119884) minus11990311990310055761005576120572120572=1

119888119888120572120572120578120578120572120572 1007665119884119884) 120578120578120572120572 1007665119884119884)10077031007703

minus 119883119883 10076811007681119892119892 10076491007649nabla11988311988311988411988411988411988411988410076651007665 + 119892119892 10076491007649119884119884119884 nabla1198831198831198841198841007665100766510076971007697

+ 11988311988311990311990310055761005576120572120572=1

119888119888120572120572 10076811007681120578120578120572120572 10076491007649nabla11988311988311988411988410076651007665 120578120578

120572120572 1007665119884119884)

+120578120578120572120572 10076491007649nabla11988311988311988411988410076651007665 120578120578120572120572 1007665119884119884)10076971007697 (45)

Now using (12) in (45) we have

10076491007649nabla119883119883119861119861110076651007665 1007665119884119884119884119884119884) = 1007665119883119883 119883 119883119883)10076871007687119892119892 1007665119884119884119884119884119884) minus11990311990310055761005576120572120572=1

119888119888120572120572120578120578120572120572 1007665119884119884) 120578120578120572120572 1007665119884119884)10077031007703

minus 119883119883 10076811007681119883119883 10076491007649119892119892 1007665119884119884119884119884119884)10076651007665 minus 119861119861 1007665119883119883119884 119884119884) 120579120579 1007665119884119884)

minus119861119861 1007665119883119883119884119884119884) 120579120579 1007665119884119884)

+ 11988311988311990311990310055761005576120572120572=1

10076821007682119892119892 10076501007650nabla119883119883119884119884119884 12057712057712057212057210076661007666 120578120578120572120572 1007665119884119884)

+119892119892 10076501007650nabla119883119883119884119884119884 12057712057712057212057210076661007666 120578120578120572120572 1007665119884119884)10076661007666

(46)

Again using (12) we have

119892119892 10076501007650nabla119883119883119884119884119884 12057712057712057212057210076661007666 = 119883119883 10076501007650119892119892 10076501007650119884119884119884 1205771205771205721205721007666100766610076661007666 + 119888119888120572120572119892119892 10076491007649119884119884119884 12060112060111988311988310076651007665

+ 119888119888120572120572119906119906 1007665119883119883) 120579120579 1007665119884119884) (47)

us from (46) and (47) the rst expression of the theoremfollows Next using (38) and (42) we obtain

10076491007649nabla119883119883119861119861210076651007665 1007665119884119884119884119884119884) =11990311990310055761005576120572120572=1

119888119888120572120572 10076811007681120578120578120572120572 10076491007649nabla11988311988311988411988410076651007665 119906119906 1007665119884119884) + 120578120578120572120572 10076491007649nabla11988311988311988411988410076651007665119906119906 1007665119884119884)

+ 120578120578120572120572 1007665119884119884) 119906119906 10076491007649nabla11988311988311988411988410076651007665

+120578120578120572120572 1007665119884119884) 119906119906 10076491007649nabla1198831198831198841198841007665100766510076971007697 (48)

Using (31) and (32) we get

120578120578120572120572 10076491007649nabla11988311988311988411988410076651007665 = 119892119892 10076491007649119884119884119884 12060112060111988311988310076651007665 + 119906119906 1007665119883119883) 120579120579 1007665119884119884) 119884 (49)

and from (2) (3) (10) (16) (17) (18) and (25) we obtain

119906119906 10076491007649nabla11988311988311988411988410076651007665 = 119861119861 10076491007649119883119883119884 12060112060111988411988410076651007665 + 120591120591 1007665119883119883) 119906119906 1007665119884119884) (50)

us by substituting (49) and (50) in (48) we obtain (44)

eorem 6 Let 1198721198721007665119888119888) be an indenite 119982119982-space form and 119872119872be a totally contact umbilical lightlike hypersurface of 1198721198721007665119888119888)en 119888119888 = minus119888119888119888 where 119888119888 = 120598119903119903

120572120572=1 119888119888120572120572 and 119883119883 satises the followingdifferential equations

120585120585 119883 119883119883 + 119883119883120591120591 1007665120585120585) minus 1198831198832 = 0119884

119875119875119883119883 119883 119883119883 + 119883119883120591120591 1007665119875119875119883119883) = 0119884(51)

for any119883119883 119883 1198831007665119883119883119872119872)

Proof Let 119872119872 be a totally contact umbilical lightlike hyper-surface of an indenite 119982119982-space form 1198721198721007665119888119888) of constant 120601120601-sectional curvature 119888119888 en using (35) we have 1198611198611007665119883119883119884 120601120601119884119884) =1198831198831198921198921007665119883119883119884 120601120601119884119884) Substituting (29) (44) in (43) we obtain

1007665119883119883 119883 119883119883)10076871007687119892119892 1007665119884119884119884119884119884) minus11990311990310055761005576120572120572=1

119888119888120572120572120578120578120572120572 1007665119884119884) 120578120578120572120572 1007665119884119884)10077031007703

minus 1007665119884119884 119883 119883119883)10076871007687119892119892 1007665119883119883119884119884119884) minus11990311990310055761005576120572120572=1

119888119888120572120572120578120578120572120572 1007665119883119883) 120578120578120572120572 1007665119884119884)10077031007703

+ 119883119883 119861119861 1007665119883119883119884119884119884) 120579120579 1007665119884119884) minus 119861119861 1007665119884119884119884119884119884) 120579120579 1007665119883119883)

+ 2119883119883 10076811007681119892119892 1007649100764912060112060111988311988311988411988411988410076651007665 + 119906119906 1007665119883119883) 120579120579 1007665119884119884)10076971007697 120578120578 1007665119884119884)

+ 119883119883 1007681100768110076491007649119892119892 10076491007649119884119884119884 12060112060111988311988310076651007665 120578120578 1007665119884119884) minus 119892119892 10076491007649119884119884119884 12060112060111988411988410076651007665 120578120578 1007665119883119883)10076971007697

+ 119883119883 10076811007681119906119906 1007665119883119883) 120578120578 1007665119884119884) minus 119906119906 1007665119884119884) 120578120578 1007665119883119883)10076971007697 120579120579 1007665119884119884)

+ 2119888119888 10076811007681119892119892 1007649100764912060112060111988311988311988411988411988410076651007665 + 119906119906 1007665119883119883) 120579120579 1007665119884119884)10076971007697 119906119906 1007665119884119884)

+ 10076811007681119883119883119892119892 10076491007649119883119883119884 12060112060111988411988410076651007665 + 120591120591 1007665119883119883) 119906119906 1007665119884119884)

minus119883119883119892119892 10076491007649119884119884119884 12060112060111988311988310076651007665 minus 120591120591 1007665119884119884) 119906119906 1007665119883119883)10076971007697 120578120578 1007665119884119884)

6 Journal of Mathematics

+ 120582120582 10076811007681119892119892 10076491007649119883119883119883 11988311988311988311988310076651007665 120578120578 (119884119884) minus 119892119892 10076491007649119884119884119883 11988311988311988311988310076651007665 120578120578 (119883119883)10076971007697

+ 10076811007681120591120591 (119883119883) 120578120578 (119884119884) minus 120591120591 (119884119884) 120578120578 (119883119883)10076971007697 119906119906 (119883119883)

+ 120598120598 10076811007681119892119892 10076491007649119883119883119883 11988311988311988311988310076651007665 119906119906 (119884119884) minus 119892119892 10076491007649119883119883119883 11988311988311988411988410076651007665 119906119906 (119883119883)10076971007697

=119888119888 minus 1205981205984

10076821007682119892119892 (119881119881119883119883119883) 119892119892 10076501007650119883119883119883 11988311988311988411988410076661007666

minus 119892119892 (119881119881119883 119884119884) 119892119892 10076501007650119883119883119883 11988311988311988311988310076661007666

+2119892119892 (119881119881119883119883119883) 119892119892 10076501007650119883119883119883 1198831198831198841198841007666100766610076981007698

+ 120591120591 (119884119884) 119861119861 (119883119883119883119883119883) minus 120591120591 (119883119883) 119861119861 (119884119884119883119883119883) 119883(52)

for any119883119883119883119884119884119883119883119883 119883 119883(119883119883119872119872) Put119883119883 = 119883119883 in (52) we get

(119883119883 120585 120582120582) 10076871007687119892119892 (119884119884119883119883119883) minus11990311990310055761005576120572120572=120572

120598120598120572120572120578120578120572120572 (119884119884) 120578120578120572120572 (119883119883)10077031007703

minus 120582120582119861119861 (119884119884119883119883119883) minus 2120582120582119906119906 (119884119884) 120578120578 (119883119883)

minus 120582120582119906119906 (119883119883) 120578120578 (119884119884) minus 2120598120598119906119906 (119884119884) 119906119906 (119883119883)

+ 120591120591 (119883119883) 119906119906 (119884119884) 120578120578 (119883119883) + 2120582120582119906119906 (119884119884) 120578120578 (119883119883)

+ 120591120591 (119883119883) 119906119906 (119883119883) 120578120578 (119884119884) minus 120598120598119906119906 (119883119883) 119906119906 (119884119884)

+ 120582120582119906119906 (119883119883) 120578120578 (119884119884)

=119888119888 minus 1205981205984

119906119906 (119884119884) 119906119906 (119883119883) + 2119906119906 (119883119883) 119906119906 (119884119884)

minus 120591120591 (119883119883) 119861119861 (119884119884119883119883119883)

(53)

Put 119884119884 = 119883119883 = 119884119884 in (53) and then using 119892119892(119884119884119883119884119884) = 119892 119906119906(119884119884) =120572 120578120578120572120572(119884119884) = 119892 we obtain

minus120582120582119861119861 (119884119884119883119884119884) minus 3120598120598 = 3(119888119888 minus 120598120598)4

minus 120591120591 (119883119883) 119861119861 (119884119884119883119884119884) 119883 (54)

since 119861119861(119884119884119883119884119884) = 119892 we get 119888119888 = minus3120598120598 Moreover by putting 119884119884 =119881119881 and119883119883 = 119884119884 in (52) we obtain

119883119883 120585 120582120582 + 120582120582120591120591 (119883119883) minus 1205821205822 = 119892 (55)

Finally putting119883119883 = 119883119883119883119883 119884119884 = 119883119883119884119884 and119883119883 = 119883119883119883119883 in (52) with119888119888 = minus3120598120598 and using that 119878119878(119883119883119872119872) is nondegenerate we obtain

119883119883119883119883 120585 120582120582 + 120582120582120591120591 (119883119883119883119883)10076551007655119883119883119884119884 minus11990311990310055761005576120572120572=120572

120578120578120572120572 (119883119883119884119884) 12057712057712057212057210076711007671

= 119883119883119884119884 120585 120582120582 + 120582120582120591120591 (119883119883119884119884)10076551007655119883119883119883119883 minus11990311990310055761005576120572120572=120572

120578120578120572120572 (119883119883119883119883) 12057712057712057212057210076711007671

(56)

Putting 119883119883119883119883 = 120577120577120572120572 in (56) we get

10076821007682120577120577120572120572 120585 120582120582 + 120582120582120591120591 10076501007650120577120577120572120572100766610076661007698100769810076551007655119883119883119884119884 minus11990311990310055761005576120572120572=120572

120578120578120572120572 (119883119883119884119884) 12057712057712057212057210076711007671 = 119892 (57)

By taking 119884119884 = 119881119881 we obtain

120577120577120572120572 120585 120582120582 + 120582120582120591120591 1007650100765012057712057712057212057210076661007666 = 119892 (58)

Since 119883119883119883119883 119883 119883(119878119878(119883119883119872119872)) therefore using (19) we can write

119883119883119883119883 = 119906119906 (119883119883119883119883)119884119884 + 119880119880 (119883119883119883119883)119881119881 +2119899119899minus410055761005576119894119894=120572

120573120573119894119894119865119865119894119894

+11990311990310055761005576120572120572=120572

120598120598120572120572120578120578120572120572 (119883119883119883119883) 120577120577120572120572

= 119883119883119883119883prime +11990311990310055761005576120572120572=120572

120598120598120572120572120578120578120572120572 (119883119883119883119883) 120577120577120572120572119883

(59)

where 119865119865119894119894120572le119894119894le2119899119899minus4 is an orthogonal basis of 119863119863119892 then using(58) we have

119883119883119883119883 120585 120582120582 + 120582120582120591120591 (119883119883119883119883) = 10076551007655119883119883119883119883prime +11990311990310055761005576120572120572=120572

120598120598120572120572120578120578120572120572 (119883119883119883119883) 12057712057712057212057210076711007671 120585 120582120582

+ 12058212058212059112059110076551007655119883119883119883119883prime +11990311990310055761005576120572120572=120572

120598120598120572120572120578120578120572120572 (119883119883119883119883) 12057712057712057212057210076711007671

= 119883119883119883119883prime 120585 120582120582 + 120582120582120591120591 10076511007651119883119883119883119883prime10076671007667 119883

(60)

which leads to get from (56)

10076831007683119883119883119883119883prime 120585 120582120582 + 120582120582120591120591 10076511007651119883119883119883119883prime1007667100766710076991007699119883119883119884119884prime = 10076831007683119883119883119884119884prime 120585 120582120582 + 120582120582120591120591 10076511007651119883119883119884119884prime1007667100766710076991007699119883119883119883119883prime(61)

Now assume that there eists a vector eld 119883119883119892 on someneighborhood of119872119872 such that 119883119883119883119883prime

119892 120585 120582120582 + 120582120582120591120591(119883119883119883119883prime119892) ne 119892 at some

point 119901119901 in the neighborhood en from (61) it is clear thatall the vectors of the ber (119878119878(119883119883119872119872) minus 120577120577120572120572)119901119901 are collinear with(119883119883119883119883prime

119892)119901119901is contradicts dim(119878119878(119883119883119872119872)minus120577120577120572120572) gt 120572is impliesthe result

From the previous mentioned theorem we have thefollowing corollary

Corollary 7 ere exist no totally contact umbilical lightlikereal hypersurfaces of indenite 119982119982-space form119872119872(119888119888) (119888119888 ne minus 3120598120598)with 120577120577120572120572 119883 119883119883119872119872

Acknowledgment

e authors would like to thank the anonymous referee forhisher valuable suggestions that helped them to improve thispaper

References

[1] B Y Chen Geometry of Submanifolds Marcel Dekker NewYork USA 1973

[2] B OrsquoNeill Semi-Riemannian Geometry with Applications toRelativity Academic Press 1983

[3] K L Duggal and A Bejancu Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications Kluwer AcademicPublishers Dordrecht e Netherlands 1996

Journal of Mathematics 7

[4] L Duggal and B Sahin ldquoLightlike submanifolds of indeniteSasakian manifoldsrdquo International Journal of Mathematics andMathematical Sciences vol 2007 Article ID 57585 21 pages2007

[5] V I Arnold ldquoContact geometry the geometrical method ofGibbss thermodynamicsrdquo in Proceedings of the Gibbs Sym-posium pp 163ndash179 American Mathematical Society NewHaven Conn USA 1989

[6] SMaclaneGeometricalMechanics II Lecture Notes Universityof Chicago Chicago Ill USA 1968

[7] V E Nazaikinskii V E Shatalov and B Y Sternin ContactGeometry and Linear Differential Equations vol 6 of DeGruyter Expositions in Mathematics Walter de Gruyter BerlinGermany 1992

[8] L Brunetti and AM Pastore ldquoCurvature of a class of indeniteglobally framed f-manifoldsrdquo Bulletin Mathematique de laSociete des Sciences Mathematiques de Roumanie vol 51 no 3pp 183ndash204 2008

[9] C Calin ldquoOn existence of degenerate hypersurfaces in Sasakianmanifoldsrdquo Arab Journal of Mathematical Sciences vol 5 pp21ndash27 1999

[10] L Brunetti and A M Pastore ldquoLightlike hypersurfaces in inde-ite 119982119982-manifoldsrdquo Differential GeometrymdashDynamical Systemsvol 12 pp 18ndash40 2010

[11] A Bejancu ldquoUmbilical semi-invariant submanifolds of aSasakian manifoldrdquo Tensor vol 37 pp 203ndash213 1982

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Journal of Mathematics 5

Hence using (40) and (41) we obtain

10076491007649nabla11988311988311986111986110076651007665 1007665119884119884119884119884119884) minus 10076491007649nabla11988411988411986111986110076651007665 1007665119883119883119884119884119884)

=119888119888 minus 1198881198884

10076821007682119892119892 1007665119881119881119884119883119883) 119892119892 10076501007650119884119884119884 12060112060111988411988410076661007666 minus 119892119892 1007665119881119881119884 119884119884) 119892119892 10076501007650119884119884119884 12060112060111988311988310076661007666

+2119892119892 1007665119881119881119884119884119884) 119892119892 10076501007650119883119883119884 1206011206011198841198841007666100766610076661007666

minus 120591120591 1007665119883119883) 119861119861 1007665119884119884119884119884119884) + 120591120591 1007665119884119884) 119861119861 1007665119883119883119884119884119884) 119884

(43)

for any119883119883119884119884119884119884119884119884 119883 1198831007665119883119883119872119872)

Lemma 5 Let 1007665119872119872119884 119892119892) be a totally contact umbilical lightlikehypersurface of an indenite 119982119982-manifold 1007665119872119872119884 119892119892) then for any119883119883119884119884119884119884119884119884 119883 1198831007665119883119883119872119872) one has

10076491007649nabla119883119883119861119861110076651007665 1007665119884119884119884119884119884) = 1007665119883119883 119883 119883119883)10076871007687119892119892 1007665119884119884119884119884119884) minus11990311990310055761005576120572120572=1

119888119888120572120572120578120578120572120572 1007665119884119884) 120578120578120572120572 1007665119884119884)10077031007703

+ 119883119883 119861119861 1007665119883119883119884 119884119884) 120579120579 1007665119884119884) + 119861119861 1007665119883119883119884119884119884) 120579120579 1007665119884119884)

+ 119883119883120578120578 1007665119884119884) 10076811007681119892119892 1007649100764912060112060111988311988311988411988411988410076651007665 + 119906119906 1007665119883119883) 120579120579 1007665119884119884)10076971007697

+ 119883119883120578120578 1007665119884119884) 10076811007681119892119892 1007649100764912060112060111988311988311988411988411988410076651007665 + 119906119906 1007665119883119883) 120579120579 1007665119884119884)10076971007697

10076491007649nabla119883119883119861119861210076651007665 1007665119884119884119884119884119884) = 119888119888119906119906 1007665119884119884) 10076811007681119892119892 10076491007649119884119884119884 12060112060111988311988310076651007665 + 119906119906 1007665119883119883) 120579120579 1007665119884119884)10076971007697

+ 119888119888119906119906 1007665119884119884) 10076811007681119892119892 10076491007649119884119884119884 12060112060111988311988310076651007665 + 119906119906 1007665119883119883) 120579120579 1007665119884119884)10076971007697

+ 120578120578 1007665119884119884) 10076811007681119861119861 10076491007649119883119883119884 12060112060111988411988410076651007665 + 120591120591 1007665119883119883) 119906119906 1007665119884119884)10076971007697

+ 120578120578 1007665119884119884) 10076811007681119861119861 10076491007649119883119883119884 12060112060111988411988410076651007665 + 120591120591 1007665119883119883) 119906119906 1007665119884119884)10076971007697 (44)

Proof By virtue of (37) and (42) we obtain

10076491007649nabla119883119883119861119861110076651007665 1007665119884119884119884119884119884) = 1007665119883119883 119883 119883119883)10076871007687119892119892 1007665119884119884119884119884119884) minus11990311990310055761005576120572120572=1

119888119888120572120572120578120578120572120572 1007665119884119884) 120578120578120572120572 1007665119884119884)10077031007703

minus 119883119883 10076811007681119892119892 10076491007649nabla11988311988311988411988411988411988411988410076651007665 + 119892119892 10076491007649119884119884119884 nabla1198831198831198841198841007665100766510076971007697

+ 11988311988311990311990310055761005576120572120572=1

119888119888120572120572 10076811007681120578120578120572120572 10076491007649nabla11988311988311988411988410076651007665 120578120578

120572120572 1007665119884119884)

+120578120578120572120572 10076491007649nabla11988311988311988411988410076651007665 120578120578120572120572 1007665119884119884)10076971007697 (45)

Now using (12) in (45) we have

10076491007649nabla119883119883119861119861110076651007665 1007665119884119884119884119884119884) = 1007665119883119883 119883 119883119883)10076871007687119892119892 1007665119884119884119884119884119884) minus11990311990310055761005576120572120572=1

119888119888120572120572120578120578120572120572 1007665119884119884) 120578120578120572120572 1007665119884119884)10077031007703

minus 119883119883 10076811007681119883119883 10076491007649119892119892 1007665119884119884119884119884119884)10076651007665 minus 119861119861 1007665119883119883119884 119884119884) 120579120579 1007665119884119884)

minus119861119861 1007665119883119883119884119884119884) 120579120579 1007665119884119884)

+ 11988311988311990311990310055761005576120572120572=1

10076821007682119892119892 10076501007650nabla119883119883119884119884119884 12057712057712057212057210076661007666 120578120578120572120572 1007665119884119884)

+119892119892 10076501007650nabla119883119883119884119884119884 12057712057712057212057210076661007666 120578120578120572120572 1007665119884119884)10076661007666

(46)

Again using (12) we have

119892119892 10076501007650nabla119883119883119884119884119884 12057712057712057212057210076661007666 = 119883119883 10076501007650119892119892 10076501007650119884119884119884 1205771205771205721205721007666100766610076661007666 + 119888119888120572120572119892119892 10076491007649119884119884119884 12060112060111988311988310076651007665

+ 119888119888120572120572119906119906 1007665119883119883) 120579120579 1007665119884119884) (47)

us from (46) and (47) the rst expression of the theoremfollows Next using (38) and (42) we obtain

10076491007649nabla119883119883119861119861210076651007665 1007665119884119884119884119884119884) =11990311990310055761005576120572120572=1

119888119888120572120572 10076811007681120578120578120572120572 10076491007649nabla11988311988311988411988410076651007665 119906119906 1007665119884119884) + 120578120578120572120572 10076491007649nabla11988311988311988411988410076651007665119906119906 1007665119884119884)

+ 120578120578120572120572 1007665119884119884) 119906119906 10076491007649nabla11988311988311988411988410076651007665

+120578120578120572120572 1007665119884119884) 119906119906 10076491007649nabla1198831198831198841198841007665100766510076971007697 (48)

Using (31) and (32) we get

120578120578120572120572 10076491007649nabla11988311988311988411988410076651007665 = 119892119892 10076491007649119884119884119884 12060112060111988311988310076651007665 + 119906119906 1007665119883119883) 120579120579 1007665119884119884) 119884 (49)

and from (2) (3) (10) (16) (17) (18) and (25) we obtain

119906119906 10076491007649nabla11988311988311988411988410076651007665 = 119861119861 10076491007649119883119883119884 12060112060111988411988410076651007665 + 120591120591 1007665119883119883) 119906119906 1007665119884119884) (50)

us by substituting (49) and (50) in (48) we obtain (44)

eorem 6 Let 1198721198721007665119888119888) be an indenite 119982119982-space form and 119872119872be a totally contact umbilical lightlike hypersurface of 1198721198721007665119888119888)en 119888119888 = minus119888119888119888 where 119888119888 = 120598119903119903

120572120572=1 119888119888120572120572 and 119883119883 satises the followingdifferential equations

120585120585 119883 119883119883 + 119883119883120591120591 1007665120585120585) minus 1198831198832 = 0119884

119875119875119883119883 119883 119883119883 + 119883119883120591120591 1007665119875119875119883119883) = 0119884(51)

for any119883119883 119883 1198831007665119883119883119872119872)

Proof Let 119872119872 be a totally contact umbilical lightlike hyper-surface of an indenite 119982119982-space form 1198721198721007665119888119888) of constant 120601120601-sectional curvature 119888119888 en using (35) we have 1198611198611007665119883119883119884 120601120601119884119884) =1198831198831198921198921007665119883119883119884 120601120601119884119884) Substituting (29) (44) in (43) we obtain

1007665119883119883 119883 119883119883)10076871007687119892119892 1007665119884119884119884119884119884) minus11990311990310055761005576120572120572=1

119888119888120572120572120578120578120572120572 1007665119884119884) 120578120578120572120572 1007665119884119884)10077031007703

minus 1007665119884119884 119883 119883119883)10076871007687119892119892 1007665119883119883119884119884119884) minus11990311990310055761005576120572120572=1

119888119888120572120572120578120578120572120572 1007665119883119883) 120578120578120572120572 1007665119884119884)10077031007703

+ 119883119883 119861119861 1007665119883119883119884119884119884) 120579120579 1007665119884119884) minus 119861119861 1007665119884119884119884119884119884) 120579120579 1007665119883119883)

+ 2119883119883 10076811007681119892119892 1007649100764912060112060111988311988311988411988411988410076651007665 + 119906119906 1007665119883119883) 120579120579 1007665119884119884)10076971007697 120578120578 1007665119884119884)

+ 119883119883 1007681100768110076491007649119892119892 10076491007649119884119884119884 12060112060111988311988310076651007665 120578120578 1007665119884119884) minus 119892119892 10076491007649119884119884119884 12060112060111988411988410076651007665 120578120578 1007665119883119883)10076971007697

+ 119883119883 10076811007681119906119906 1007665119883119883) 120578120578 1007665119884119884) minus 119906119906 1007665119884119884) 120578120578 1007665119883119883)10076971007697 120579120579 1007665119884119884)

+ 2119888119888 10076811007681119892119892 1007649100764912060112060111988311988311988411988411988410076651007665 + 119906119906 1007665119883119883) 120579120579 1007665119884119884)10076971007697 119906119906 1007665119884119884)

+ 10076811007681119883119883119892119892 10076491007649119883119883119884 12060112060111988411988410076651007665 + 120591120591 1007665119883119883) 119906119906 1007665119884119884)

minus119883119883119892119892 10076491007649119884119884119884 12060112060111988311988310076651007665 minus 120591120591 1007665119884119884) 119906119906 1007665119883119883)10076971007697 120578120578 1007665119884119884)

6 Journal of Mathematics

+ 120582120582 10076811007681119892119892 10076491007649119883119883119883 11988311988311988311988310076651007665 120578120578 (119884119884) minus 119892119892 10076491007649119884119884119883 11988311988311988311988310076651007665 120578120578 (119883119883)10076971007697

+ 10076811007681120591120591 (119883119883) 120578120578 (119884119884) minus 120591120591 (119884119884) 120578120578 (119883119883)10076971007697 119906119906 (119883119883)

+ 120598120598 10076811007681119892119892 10076491007649119883119883119883 11988311988311988311988310076651007665 119906119906 (119884119884) minus 119892119892 10076491007649119883119883119883 11988311988311988411988410076651007665 119906119906 (119883119883)10076971007697

=119888119888 minus 1205981205984

10076821007682119892119892 (119881119881119883119883119883) 119892119892 10076501007650119883119883119883 11988311988311988411988410076661007666

minus 119892119892 (119881119881119883 119884119884) 119892119892 10076501007650119883119883119883 11988311988311988311988310076661007666

+2119892119892 (119881119881119883119883119883) 119892119892 10076501007650119883119883119883 1198831198831198841198841007666100766610076981007698

+ 120591120591 (119884119884) 119861119861 (119883119883119883119883119883) minus 120591120591 (119883119883) 119861119861 (119884119884119883119883119883) 119883(52)

for any119883119883119883119884119884119883119883119883 119883 119883(119883119883119872119872) Put119883119883 = 119883119883 in (52) we get

(119883119883 120585 120582120582) 10076871007687119892119892 (119884119884119883119883119883) minus11990311990310055761005576120572120572=120572

120598120598120572120572120578120578120572120572 (119884119884) 120578120578120572120572 (119883119883)10077031007703

minus 120582120582119861119861 (119884119884119883119883119883) minus 2120582120582119906119906 (119884119884) 120578120578 (119883119883)

minus 120582120582119906119906 (119883119883) 120578120578 (119884119884) minus 2120598120598119906119906 (119884119884) 119906119906 (119883119883)

+ 120591120591 (119883119883) 119906119906 (119884119884) 120578120578 (119883119883) + 2120582120582119906119906 (119884119884) 120578120578 (119883119883)

+ 120591120591 (119883119883) 119906119906 (119883119883) 120578120578 (119884119884) minus 120598120598119906119906 (119883119883) 119906119906 (119884119884)

+ 120582120582119906119906 (119883119883) 120578120578 (119884119884)

=119888119888 minus 1205981205984

119906119906 (119884119884) 119906119906 (119883119883) + 2119906119906 (119883119883) 119906119906 (119884119884)

minus 120591120591 (119883119883) 119861119861 (119884119884119883119883119883)

(53)

Put 119884119884 = 119883119883 = 119884119884 in (53) and then using 119892119892(119884119884119883119884119884) = 119892 119906119906(119884119884) =120572 120578120578120572120572(119884119884) = 119892 we obtain

minus120582120582119861119861 (119884119884119883119884119884) minus 3120598120598 = 3(119888119888 minus 120598120598)4

minus 120591120591 (119883119883) 119861119861 (119884119884119883119884119884) 119883 (54)

since 119861119861(119884119884119883119884119884) = 119892 we get 119888119888 = minus3120598120598 Moreover by putting 119884119884 =119881119881 and119883119883 = 119884119884 in (52) we obtain

119883119883 120585 120582120582 + 120582120582120591120591 (119883119883) minus 1205821205822 = 119892 (55)

Finally putting119883119883 = 119883119883119883119883 119884119884 = 119883119883119884119884 and119883119883 = 119883119883119883119883 in (52) with119888119888 = minus3120598120598 and using that 119878119878(119883119883119872119872) is nondegenerate we obtain

119883119883119883119883 120585 120582120582 + 120582120582120591120591 (119883119883119883119883)10076551007655119883119883119884119884 minus11990311990310055761005576120572120572=120572

120578120578120572120572 (119883119883119884119884) 12057712057712057212057210076711007671

= 119883119883119884119884 120585 120582120582 + 120582120582120591120591 (119883119883119884119884)10076551007655119883119883119883119883 minus11990311990310055761005576120572120572=120572

120578120578120572120572 (119883119883119883119883) 12057712057712057212057210076711007671

(56)

Putting 119883119883119883119883 = 120577120577120572120572 in (56) we get

10076821007682120577120577120572120572 120585 120582120582 + 120582120582120591120591 10076501007650120577120577120572120572100766610076661007698100769810076551007655119883119883119884119884 minus11990311990310055761005576120572120572=120572

120578120578120572120572 (119883119883119884119884) 12057712057712057212057210076711007671 = 119892 (57)

By taking 119884119884 = 119881119881 we obtain

120577120577120572120572 120585 120582120582 + 120582120582120591120591 1007650100765012057712057712057212057210076661007666 = 119892 (58)

Since 119883119883119883119883 119883 119883(119878119878(119883119883119872119872)) therefore using (19) we can write

119883119883119883119883 = 119906119906 (119883119883119883119883)119884119884 + 119880119880 (119883119883119883119883)119881119881 +2119899119899minus410055761005576119894119894=120572

120573120573119894119894119865119865119894119894

+11990311990310055761005576120572120572=120572

120598120598120572120572120578120578120572120572 (119883119883119883119883) 120577120577120572120572

= 119883119883119883119883prime +11990311990310055761005576120572120572=120572

120598120598120572120572120578120578120572120572 (119883119883119883119883) 120577120577120572120572119883

(59)

where 119865119865119894119894120572le119894119894le2119899119899minus4 is an orthogonal basis of 119863119863119892 then using(58) we have

119883119883119883119883 120585 120582120582 + 120582120582120591120591 (119883119883119883119883) = 10076551007655119883119883119883119883prime +11990311990310055761005576120572120572=120572

120598120598120572120572120578120578120572120572 (119883119883119883119883) 12057712057712057212057210076711007671 120585 120582120582

+ 12058212058212059112059110076551007655119883119883119883119883prime +11990311990310055761005576120572120572=120572

120598120598120572120572120578120578120572120572 (119883119883119883119883) 12057712057712057212057210076711007671

= 119883119883119883119883prime 120585 120582120582 + 120582120582120591120591 10076511007651119883119883119883119883prime10076671007667 119883

(60)

which leads to get from (56)

10076831007683119883119883119883119883prime 120585 120582120582 + 120582120582120591120591 10076511007651119883119883119883119883prime1007667100766710076991007699119883119883119884119884prime = 10076831007683119883119883119884119884prime 120585 120582120582 + 120582120582120591120591 10076511007651119883119883119884119884prime1007667100766710076991007699119883119883119883119883prime(61)

Now assume that there eists a vector eld 119883119883119892 on someneighborhood of119872119872 such that 119883119883119883119883prime

119892 120585 120582120582 + 120582120582120591120591(119883119883119883119883prime119892) ne 119892 at some

point 119901119901 in the neighborhood en from (61) it is clear thatall the vectors of the ber (119878119878(119883119883119872119872) minus 120577120577120572120572)119901119901 are collinear with(119883119883119883119883prime

119892)119901119901is contradicts dim(119878119878(119883119883119872119872)minus120577120577120572120572) gt 120572is impliesthe result

From the previous mentioned theorem we have thefollowing corollary

Corollary 7 ere exist no totally contact umbilical lightlikereal hypersurfaces of indenite 119982119982-space form119872119872(119888119888) (119888119888 ne minus 3120598120598)with 120577120577120572120572 119883 119883119883119872119872

Acknowledgment

e authors would like to thank the anonymous referee forhisher valuable suggestions that helped them to improve thispaper

References

[1] B Y Chen Geometry of Submanifolds Marcel Dekker NewYork USA 1973

[2] B OrsquoNeill Semi-Riemannian Geometry with Applications toRelativity Academic Press 1983

[3] K L Duggal and A Bejancu Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications Kluwer AcademicPublishers Dordrecht e Netherlands 1996

Journal of Mathematics 7

[4] L Duggal and B Sahin ldquoLightlike submanifolds of indeniteSasakian manifoldsrdquo International Journal of Mathematics andMathematical Sciences vol 2007 Article ID 57585 21 pages2007

[5] V I Arnold ldquoContact geometry the geometrical method ofGibbss thermodynamicsrdquo in Proceedings of the Gibbs Sym-posium pp 163ndash179 American Mathematical Society NewHaven Conn USA 1989

[6] SMaclaneGeometricalMechanics II Lecture Notes Universityof Chicago Chicago Ill USA 1968

[7] V E Nazaikinskii V E Shatalov and B Y Sternin ContactGeometry and Linear Differential Equations vol 6 of DeGruyter Expositions in Mathematics Walter de Gruyter BerlinGermany 1992

[8] L Brunetti and AM Pastore ldquoCurvature of a class of indeniteglobally framed f-manifoldsrdquo Bulletin Mathematique de laSociete des Sciences Mathematiques de Roumanie vol 51 no 3pp 183ndash204 2008

[9] C Calin ldquoOn existence of degenerate hypersurfaces in Sasakianmanifoldsrdquo Arab Journal of Mathematical Sciences vol 5 pp21ndash27 1999

[10] L Brunetti and A M Pastore ldquoLightlike hypersurfaces in inde-ite 119982119982-manifoldsrdquo Differential GeometrymdashDynamical Systemsvol 12 pp 18ndash40 2010

[11] A Bejancu ldquoUmbilical semi-invariant submanifolds of aSasakian manifoldrdquo Tensor vol 37 pp 203ndash213 1982

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 Journal of Mathematics

+ 120582120582 10076811007681119892119892 10076491007649119883119883119883 11988311988311988311988310076651007665 120578120578 (119884119884) minus 119892119892 10076491007649119884119884119883 11988311988311988311988310076651007665 120578120578 (119883119883)10076971007697

+ 10076811007681120591120591 (119883119883) 120578120578 (119884119884) minus 120591120591 (119884119884) 120578120578 (119883119883)10076971007697 119906119906 (119883119883)

+ 120598120598 10076811007681119892119892 10076491007649119883119883119883 11988311988311988311988310076651007665 119906119906 (119884119884) minus 119892119892 10076491007649119883119883119883 11988311988311988411988410076651007665 119906119906 (119883119883)10076971007697

=119888119888 minus 1205981205984

10076821007682119892119892 (119881119881119883119883119883) 119892119892 10076501007650119883119883119883 11988311988311988411988410076661007666

minus 119892119892 (119881119881119883 119884119884) 119892119892 10076501007650119883119883119883 11988311988311988311988310076661007666

+2119892119892 (119881119881119883119883119883) 119892119892 10076501007650119883119883119883 1198831198831198841198841007666100766610076981007698

+ 120591120591 (119884119884) 119861119861 (119883119883119883119883119883) minus 120591120591 (119883119883) 119861119861 (119884119884119883119883119883) 119883(52)

for any119883119883119883119884119884119883119883119883 119883 119883(119883119883119872119872) Put119883119883 = 119883119883 in (52) we get

(119883119883 120585 120582120582) 10076871007687119892119892 (119884119884119883119883119883) minus11990311990310055761005576120572120572=120572

120598120598120572120572120578120578120572120572 (119884119884) 120578120578120572120572 (119883119883)10077031007703

minus 120582120582119861119861 (119884119884119883119883119883) minus 2120582120582119906119906 (119884119884) 120578120578 (119883119883)

minus 120582120582119906119906 (119883119883) 120578120578 (119884119884) minus 2120598120598119906119906 (119884119884) 119906119906 (119883119883)

+ 120591120591 (119883119883) 119906119906 (119884119884) 120578120578 (119883119883) + 2120582120582119906119906 (119884119884) 120578120578 (119883119883)

+ 120591120591 (119883119883) 119906119906 (119883119883) 120578120578 (119884119884) minus 120598120598119906119906 (119883119883) 119906119906 (119884119884)

+ 120582120582119906119906 (119883119883) 120578120578 (119884119884)

=119888119888 minus 1205981205984

119906119906 (119884119884) 119906119906 (119883119883) + 2119906119906 (119883119883) 119906119906 (119884119884)

minus 120591120591 (119883119883) 119861119861 (119884119884119883119883119883)

(53)

Put 119884119884 = 119883119883 = 119884119884 in (53) and then using 119892119892(119884119884119883119884119884) = 119892 119906119906(119884119884) =120572 120578120578120572120572(119884119884) = 119892 we obtain

minus120582120582119861119861 (119884119884119883119884119884) minus 3120598120598 = 3(119888119888 minus 120598120598)4

minus 120591120591 (119883119883) 119861119861 (119884119884119883119884119884) 119883 (54)

since 119861119861(119884119884119883119884119884) = 119892 we get 119888119888 = minus3120598120598 Moreover by putting 119884119884 =119881119881 and119883119883 = 119884119884 in (52) we obtain

119883119883 120585 120582120582 + 120582120582120591120591 (119883119883) minus 1205821205822 = 119892 (55)

Finally putting119883119883 = 119883119883119883119883 119884119884 = 119883119883119884119884 and119883119883 = 119883119883119883119883 in (52) with119888119888 = minus3120598120598 and using that 119878119878(119883119883119872119872) is nondegenerate we obtain

119883119883119883119883 120585 120582120582 + 120582120582120591120591 (119883119883119883119883)10076551007655119883119883119884119884 minus11990311990310055761005576120572120572=120572

120578120578120572120572 (119883119883119884119884) 12057712057712057212057210076711007671

= 119883119883119884119884 120585 120582120582 + 120582120582120591120591 (119883119883119884119884)10076551007655119883119883119883119883 minus11990311990310055761005576120572120572=120572

120578120578120572120572 (119883119883119883119883) 12057712057712057212057210076711007671

(56)

Putting 119883119883119883119883 = 120577120577120572120572 in (56) we get

10076821007682120577120577120572120572 120585 120582120582 + 120582120582120591120591 10076501007650120577120577120572120572100766610076661007698100769810076551007655119883119883119884119884 minus11990311990310055761005576120572120572=120572

120578120578120572120572 (119883119883119884119884) 12057712057712057212057210076711007671 = 119892 (57)

By taking 119884119884 = 119881119881 we obtain

120577120577120572120572 120585 120582120582 + 120582120582120591120591 1007650100765012057712057712057212057210076661007666 = 119892 (58)

Since 119883119883119883119883 119883 119883(119878119878(119883119883119872119872)) therefore using (19) we can write

119883119883119883119883 = 119906119906 (119883119883119883119883)119884119884 + 119880119880 (119883119883119883119883)119881119881 +2119899119899minus410055761005576119894119894=120572

120573120573119894119894119865119865119894119894

+11990311990310055761005576120572120572=120572

120598120598120572120572120578120578120572120572 (119883119883119883119883) 120577120577120572120572

= 119883119883119883119883prime +11990311990310055761005576120572120572=120572

120598120598120572120572120578120578120572120572 (119883119883119883119883) 120577120577120572120572119883

(59)

where 119865119865119894119894120572le119894119894le2119899119899minus4 is an orthogonal basis of 119863119863119892 then using(58) we have

119883119883119883119883 120585 120582120582 + 120582120582120591120591 (119883119883119883119883) = 10076551007655119883119883119883119883prime +11990311990310055761005576120572120572=120572

120598120598120572120572120578120578120572120572 (119883119883119883119883) 12057712057712057212057210076711007671 120585 120582120582

+ 12058212058212059112059110076551007655119883119883119883119883prime +11990311990310055761005576120572120572=120572

120598120598120572120572120578120578120572120572 (119883119883119883119883) 12057712057712057212057210076711007671

= 119883119883119883119883prime 120585 120582120582 + 120582120582120591120591 10076511007651119883119883119883119883prime10076671007667 119883

(60)

which leads to get from (56)

10076831007683119883119883119883119883prime 120585 120582120582 + 120582120582120591120591 10076511007651119883119883119883119883prime1007667100766710076991007699119883119883119884119884prime = 10076831007683119883119883119884119884prime 120585 120582120582 + 120582120582120591120591 10076511007651119883119883119884119884prime1007667100766710076991007699119883119883119883119883prime(61)

Now assume that there eists a vector eld 119883119883119892 on someneighborhood of119872119872 such that 119883119883119883119883prime

119892 120585 120582120582 + 120582120582120591120591(119883119883119883119883prime119892) ne 119892 at some

point 119901119901 in the neighborhood en from (61) it is clear thatall the vectors of the ber (119878119878(119883119883119872119872) minus 120577120577120572120572)119901119901 are collinear with(119883119883119883119883prime

119892)119901119901is contradicts dim(119878119878(119883119883119872119872)minus120577120577120572120572) gt 120572is impliesthe result

From the previous mentioned theorem we have thefollowing corollary

Corollary 7 ere exist no totally contact umbilical lightlikereal hypersurfaces of indenite 119982119982-space form119872119872(119888119888) (119888119888 ne minus 3120598120598)with 120577120577120572120572 119883 119883119883119872119872

Acknowledgment

e authors would like to thank the anonymous referee forhisher valuable suggestions that helped them to improve thispaper

References

[1] B Y Chen Geometry of Submanifolds Marcel Dekker NewYork USA 1973

[2] B OrsquoNeill Semi-Riemannian Geometry with Applications toRelativity Academic Press 1983

[3] K L Duggal and A Bejancu Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications Kluwer AcademicPublishers Dordrecht e Netherlands 1996

Journal of Mathematics 7

[4] L Duggal and B Sahin ldquoLightlike submanifolds of indeniteSasakian manifoldsrdquo International Journal of Mathematics andMathematical Sciences vol 2007 Article ID 57585 21 pages2007

[5] V I Arnold ldquoContact geometry the geometrical method ofGibbss thermodynamicsrdquo in Proceedings of the Gibbs Sym-posium pp 163ndash179 American Mathematical Society NewHaven Conn USA 1989

[6] SMaclaneGeometricalMechanics II Lecture Notes Universityof Chicago Chicago Ill USA 1968

[7] V E Nazaikinskii V E Shatalov and B Y Sternin ContactGeometry and Linear Differential Equations vol 6 of DeGruyter Expositions in Mathematics Walter de Gruyter BerlinGermany 1992

[8] L Brunetti and AM Pastore ldquoCurvature of a class of indeniteglobally framed f-manifoldsrdquo Bulletin Mathematique de laSociete des Sciences Mathematiques de Roumanie vol 51 no 3pp 183ndash204 2008

[9] C Calin ldquoOn existence of degenerate hypersurfaces in Sasakianmanifoldsrdquo Arab Journal of Mathematical Sciences vol 5 pp21ndash27 1999

[10] L Brunetti and A M Pastore ldquoLightlike hypersurfaces in inde-ite 119982119982-manifoldsrdquo Differential GeometrymdashDynamical Systemsvol 12 pp 18ndash40 2010

[11] A Bejancu ldquoUmbilical semi-invariant submanifolds of aSasakian manifoldrdquo Tensor vol 37 pp 203ndash213 1982

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

Journal of Mathematics 7

[4] L Duggal and B Sahin ldquoLightlike submanifolds of indeniteSasakian manifoldsrdquo International Journal of Mathematics andMathematical Sciences vol 2007 Article ID 57585 21 pages2007

[5] V I Arnold ldquoContact geometry the geometrical method ofGibbss thermodynamicsrdquo in Proceedings of the Gibbs Sym-posium pp 163ndash179 American Mathematical Society NewHaven Conn USA 1989

[6] SMaclaneGeometricalMechanics II Lecture Notes Universityof Chicago Chicago Ill USA 1968

[7] V E Nazaikinskii V E Shatalov and B Y Sternin ContactGeometry and Linear Differential Equations vol 6 of DeGruyter Expositions in Mathematics Walter de Gruyter BerlinGermany 1992

[8] L Brunetti and AM Pastore ldquoCurvature of a class of indeniteglobally framed f-manifoldsrdquo Bulletin Mathematique de laSociete des Sciences Mathematiques de Roumanie vol 51 no 3pp 183ndash204 2008

[9] C Calin ldquoOn existence of degenerate hypersurfaces in Sasakianmanifoldsrdquo Arab Journal of Mathematical Sciences vol 5 pp21ndash27 1999

[10] L Brunetti and A M Pastore ldquoLightlike hypersurfaces in inde-ite 119982119982-manifoldsrdquo Differential GeometrymdashDynamical Systemsvol 12 pp 18ndash40 2010

[11] A Bejancu ldquoUmbilical semi-invariant submanifolds of aSasakian manifoldrdquo Tensor vol 37 pp 203ndash213 1982

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