# GCR-Lightlike Product of Indefinite Sasakian Manifolds

Authors: Rakesh Kumar; Varun Jain; R. K. Nagaich
Journal: Advances in Mathematical Physics
 (2011)

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## Abstract

We study mixed geodesicGCR-lightlike submanifolds of indefinite Sasakian manifolds and obtain some necessary and sufficient conditions for a GCR-lightlike submanifold to be a GCR-lightlike product.


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## Body

## 1. Introduction

The geometry of submanifolds is one of the most important topics of differential geometry. It is well known that the geometry of semi-Riemannian submanifolds have many similarities with their Riemannian case but the geometry of lightlike submanifolds is different since their normal vector bundle intersect with the tangent bundle making it more difficult and interesting to study. The lightlike submanifolds of semi-Riemannian manifolds were introduced and studied by Duggal and Bejancu [1]. Since contact geometry has vital role in the theory of differential equations, optics, and phase spaces of a dynamical system, therefore contact geometry with definite and indefinite metric becomes the topic of main discussion. In the process of establishment of theory of lightlike submanifolds, Duggal and Sahin [2] introduced the theory of contact CR-lightlike submanifold of indefinite Sasakian manifold. Later on, Duggal and Sahin [3] introduced generalized Cauchy-Riemann (GCR)-lightlike submanifold of indefinite Sasakian manifolds to find an umbrella of invariant, screen, real, contact CR-lightlike subcases, and real hypersurfaces. In the present paper we study GCR-lightlike submanifolds extensively and study mixed geodesic GCR-lightlike submanifolds of indefinite Sasakian manifolds. We also obtain some necessary and sufficient conditions for a GCR-lightlike submanifold to be a GCR-lightlike product.


## 2. Lightlike Submanifolds

We recall notations and fundamental equations for lightlike submanifolds, which are due to the book [1] by Duggal and Bejancu.Let(M¯,g¯) be a real (m+n)-dimensional semi-Riemannian manifold of constant index q such that m,n≥1, 1≤q≤m+n-1 and (M,g) be an m-dimensional submanifold of M¯ and g be the induced metric of g¯ on M. If g¯ is degenerate on the tangent bundle TM of M then M is called a lightlike submanifold of M¯. For a degenerate metric g on M(2.1)TM⊥=∪{u∈TxM¯:g¯(u,v)=0,∀v∈TxM,x∈M},
is a degenerate n-dimensional subspace of TxM¯. Thus both TxM and TxM⊥ are degenerate orthogonal subspaces but no longer complementary. In this case, there exists a subspace RadTxM=TxM∩TxM⊥ which is known as radical (null) subspace. If the mapping(2.2)RadTM:x∈M⟶RadTxM,
defines a smooth distribution on M of rank r>0 then the submanifold M of M¯ is called an r-lightlike submanifold and RadTM is called the radical distribution on M.LetS(TM) be a screen distribution which is a semi-Riemannian complementary distribution of Rad(TM) in TM, that is(2.3)TM=RadTM⊥S(TM),
and S(TM⊥) is a complementary vector subbundle to RadTM in TM⊥. Let tr(TM) and ltr(TM) be complementary (but not orthogonal) vector bundles to TM in TM¯|M and to RadTM in S(TM⊥)⊥, respectively. Then, we have
(2.4)tr⁡(TM)=ltr(TM)⊥S(TM⊥),(2.5)TM¯|M=TM⊕tr⁡(TM)=(RadTM⊕ltr(TM))⊥S(TM)⊥S(TM⊥).Letu be a local coordinate neighborhood of M and consider the local quasiorthonormal fields of frames of M¯ along M, on u as {ξ1,…,ξr,Wr+1,…,Wn,N1,…,Nr,Xr+1,…,Xm}, where {ξ1,…,ξr},{N1,…,Nr} are local lightlike bases of Γ(RadTM|u), Γ(ltr(TM)|u) and {Wr+1,…,Wn},{Xr+1,…,Xm} are local orthonormal bases of Γ(S(TM⊥)|u) and Γ(S(TM)|u), respectively. For this quasi-orthonormal fields of frames, we have the following.Theorem 2.1 (see [1]).
Let(M,g,S(TM),S(TM⊥)) be an r-lightlike submanifold of a semi-Riemannian manifold (M¯,g¯). Then there exists a complementary vector bundle ltr(TM) of RadTM in S(TM⊥)⊥ and a basis of Γ(ltr(TM)|u) consisting of smooth section {Ni} of S(TM⊥)⊥|u, where u is a coordinate neighborhood of M such that
(2.6)g¯(Ni,ξj)=δij,g¯(Ni,Nj)=0,foranyi,j∈{1,2,…,r},
where {ξ1,…,ξr} is a lightlike basis of Γ(Rad(TM)).Let∇¯ be the Levi-Civita connection on M¯. Then according to the decomposition (2.5), the Gauss and Weingarten formulas are given by
(2.7)∇¯XY=∇XY+h(X,Y),∀X,Y∈Γ(TM),(2.8)∇¯XU=-AUX+∇X⊥U,∀X∈Γ(TM),U∈Γ(tr⁡(TM)),where{∇XY,AUX} and {h(X,Y),∇X⊥U} belong to Γ(TM) and Γ(tr(TM)), respectively. Here ∇ is a torsion-free linear connection on M, h is a symmetric bilinear form on Γ(TM) which is called the second fundamental form, and AU is linear a operator on M and called a shape operator.According to (2.4), considering the projection morphisms L and S of tr(TM) on ltr(TM) and S(TM⊥), respectively, (2.11) and (2.12) give
(2.9)∇¯XY=∇XY+hl(X,Y)+hs(X,Y),(2.10)∇¯XU=-AUX+DXlU+DXsU,
where one puts hl(X,Y)=L(h(X,Y)),hs(X,Y)=S(h(X,Y)),DXlU=L(∇X⊥U), DXsU=S(∇X⊥U).Ashl and hs are Γ(ltr(TM))-valued and Γ(S(TM⊥))-valued, respectively, therefore, these are called as the lightlike second fundamental form and the screen second fundamental form on M. In particular
(2.11)∇¯XN=-ANX+∇XlN+Ds(X,N),(2.12)∇¯XW=-AWX+∇XsW+Dl(X,W),
where X∈Γ(TM),N∈Γ(ltr(TM)) and W∈Γ(S(TM⊥)). Using (2.4)-(2.5) and (2.8)–(2.12), one obtains
(2.13)g¯(hs(X,Y),W)+g¯(Y,Dl(X,W))=g(AWX,Y),(2.14)g¯(hl(X,Y),ξ)+g¯(Y,hl(X,ξ))+g(Y,∇Xξ)=0,(2.15)g¯(ANX,N′)+g¯(N,AN′X)=0,
for any ξ∈Γ(RadTM), W∈Γ(S(TM⊥)) and N,N′∈Γ(ltr(TM)).LetP be the projection morphism of TM on S(TM). Then using (2.3), one can induce some new geometric objects on the screen distribution S(TM) on M as
(2.16)∇XPY=∇X*PY+h*(X,Y),(2.17)∇Xξ=-Aξ*X+∇X*tξ,
for any X,Y∈Γ(TM) and ξ∈Γ(RadTM), where {∇X*PY,Aξ*X} and {h*(X,Y),∇X*tξ} belong to Γ(S(TM)) and Γ(RadTM), respectively. ∇* and ∇*t are linear connections on complementary distributions S(TM) and RadTM, respectively. h* and A* are Γ(RadTM)-valued and Γ(S(TM))-valued bilinear forms and called as the second fundamental forms of distributions S(TM) and RadTM, respectively, and one has the following equations:
(2.18)g¯(hl(X,PY),ξ)=g(Aξ*X,PY),g¯(h*(X,PY),N)=g(ANX,PY)(2.19)g(Aξ*PX,PY)=g(PX,Aξ*PY),Aξ*ξ=0.Next, one recalls some basic definition and results of indefinite Sasakian manifolds [4]. An odd dimensional semi-Riemannian manifolds (M¯,g¯) is called an contact metric manifold, if there is a (1,1) tensor field ϕ, a vector field V called characteristic vector field, and a 1-form η such that
(2.20)g¯(ϕX,ϕY)=g¯(X,Y)-ϵη(X)η(Y),g¯(V,V)=ϵ,(2.21)ϕ2(X)=-X+η(X)V,g¯(X,V)=ϵη(X),(2.22)dη(X,Y)=g¯(X,ϕY),∀X,Y∈Γ(TM),
where ϵ=±1. Therefore it follows that
(2.23)ϕV=0,ηoϕ=0,η(V)=1.
Then (ϕ,V,η,g¯) is called contact metric structure of M¯. one says that M¯ has a normal contact structure if Nϕ+dη⊗V=0, where Nϕ is the Nijenhuis tensor field then M¯ is called an indefinite Sasakian manifold and for which one has
(2.24)∇¯XV=-ϕX.(2.25)(∇¯Xϕ)Y=-g¯(X,Y)V+ϵη(Y)X.


## 3. Generalized Cauchy-Riemann- (GCR-) Lightlike Submanifold

Calin [5] proved that if the characteristic vector field V is tangent to (M,g,S(TM)) then it belongs to S(TM). We assume characteristic vector V is tangent to M, throughout this paper.Definition 3.1.
Let(M,g,S(TM)) be a real lightlike submanifold of an indefinite Sasakian manifold (M¯,g¯) then M is called generalized Cauchy-Riemann (GCR)-lightlike submanifold if the following conditions are satisfied.  (A)
There exist two subbundlesD1 and D2 of Rad(TM) such that(3.1)Rad(TM)=D1⊕D2,ϕ(D1)=D1,ϕ(D2)⊂S(TM).(B)
There exist two subbundlesD0 and D¯ of S(TM) such that(3.2)S(TM)={ϕD2⊕D¯}⊥D0⊥V,ϕ(D¯)=L⊥S,
where D0 is invariant nondegenerate distribution on M, {V} is one dimensional distribution spanned by V, and L, S are vector subbundles of ltr(TM) and S(TM)⊥, respectively.Then tangent bundleTM of M is decomposed as(3.3)TM={D⊕D¯⊕{V}},D=Rad(TM)⊕D0⊕ϕ(D2).
Let Q, P1, P2 be the projection morphism on D, ϕS, ϕL, respectively, therefore(3.4)X=QX+V+P1X+P2X,
for X∈Γ(TM). Applying ϕ to (3.4), we obtain(3.5)ϕX=fX+ωP1X+ωP2X,
where fX∈Γ(D), ωP1X∈Γ(S), and ωP2X∈Γ(L) and we can write (3.5) as(3.6)ϕX=fX+ωX,
where fX and ωX are the tangential and transversal components of ϕX, respectively. Similarly(3.7)ϕU=BU+CU,U∈Γ(tr⁡(TM)),
where BU and CU are the sections of TM and tr(TM), respectively.Differentiating (3.5) and using (2.11)–(2.13), (2.14), and (3.7), we have
(3.8)Dl(X,ωP1Y)=-∇XlωP2Y+ωP2∇XY-hl(X,fY)+Chl(X,Y),Ds(X,ωP2Y)=-∇XsωP1Y+ωP1∇XY-hs(X,fY)+Chs(X,Y),
for all X,Y∈Γ(TM). Using Sasakian property of ∇¯ with (2.13) and (2.14), we have the following lemmas.Lemma 3.2.
LetM be a GCR-lightlike submanifold of an indefinite Sasakian manifold M¯. Then one has
(3.9)(∇Xf)Y=AωYX+Bh(X,Y)-g(X,Y)V+ϵη(Y)X,(3.10)(∇Xtω)Y=Ch(X,Y)-h(X,fY),
where X,Y∈Γ(TM) and
(3.11)(∇Xf)Y=∇XfY-f∇XY,(3.12)(∇Xtω)Y=∇XtωY-ω∇XY.Lemma 3.3.
LetM be a GCR-lightlike submanifold of an indefinite Sasakian manifold M¯. Then one has
(3.13)(∇XB)U=ACUX-fAUX,(∇XtC)U=-ωAUX-h(X,BU),
where X∈Γ(TM), U∈Γ(tr(TM)) and
(3.14)(∇XB)U=∇XBU-B∇XtU,(∇XtC)U=∇XtCU-C∇XtU.


## 4. Mixed Geodesic GCR-Lightlike Submanifolds

Definition 4.1.
AGCR-lightlike submanifold of an indefinite Sasakian manifold is called mixed geodesic GCR-lightlike submanifold if its second fundamental form h satisfies h(X,Y)=0 for any X∈Γ(D) and Y∈Γ(D̅).Definition 4.2.
AGCR-lightlike submanifold of an indefinite Sasakian manifold is called D′ geodesic GCR-lightlike submanifold if its second fundamental form h satisfies h(X,Y)=0 for any X,Y∈Γ(D̅).Theorem 4.3.
LetM be a GCR-lightlike submanifold of an indefinite Sasakian manifold M¯. Then M is mixed geodesic if and only if Aξ*X and AWX∉Γ(M2⊥ϕD2), for any X∈Γ(D⊕V),W∈Γ(S(TM⊥)) and ξ∈Γ(Rad(TM)).Proof.
Using definition ofGCR-lightlike submanifolds, M is mixed geodesic if and only if, g¯(h(X,Y),W)=g¯(h(X,Y),ξ)=0 for X∈Γ(D⊕V),Y∈Γ(D¯),W∈Γ(S(TM⊥)), and ξ∈Γ(Rad(TM)). Using (2.12) and (2.17) we get
(4.1)g¯(h(X,Y),W)=g¯(∇¯XY,W)=-g¯(Y,∇¯XW)=g(Y,AWX),g¯(h(X,Y),ξ)=g¯(∇¯XY,ξ)=-g(Y,∇Xξ)=g(Y,Aξ*X).
Therefore from (4.1), the proof is complete.Theorem 4.4.
LetM be a GCR-lightlike submanifold of an indefinite Sasakian manifold M¯. Then M is D¯ geodesic if and only if Aξ*X and AWX∉Γ(M2⊥ϕD2), for any X∈Γ(D¯),ξ∈ΓRad(TM) and W∈Γ(S(TM⊥)).Proof.
Proof is similar to the proof of Theorem4.3.Lemma 4.5.
LetM be a mixed geodesic GCR-lightlike submanifold of an indefinite Sasakian manifold M¯. Then Aξ*X∈Γ(ϕD2), for any X∈Γ(D¯), ξ∈Γ(D2).Proof.
ForX∈Γ(D¯) and ξ∈Γ(D2), using (2.9) we have
(4.2)h(ϕξ,X)=∇¯Xϕξ-∇Xϕξ=(∇¯Xϕ)ξ+ϕ∇¯Xξ-∇Xϕξ=-g(X,ξ)V+ϵη(ξ)X+ϕ∇¯Xξ-∇Xϕξ=ϕ∇Xξ+ϕh(X,ξ)-∇Xϕξ.
Since M is mixed geodesic therefore ϕ∇Xξ=∇Xϕξ. Using (2.16) and (2.17) we get ϕ(-Aξ*X+∇X*tξ)=∇X*ϕξ+h*(X,ϕξ) then using (3.6) we obtain -fAξ*X-ωAξ*X+ϕ(∇X*tξ)=∇X*ϕξ+h*(X,ϕξ). Comparing the transversal components we get
(4.3)ωAξ*X=0,
or
(4.4)Aξ*X∈Γ(D0⊕{V}⊥ϕ(D2)).
If Aξ*X∈D0 then the nondegeneracy of D0 implies that there must exists a Z0∈D0 such that
(4.5)g¯(Aξ*X,Z0)≠0.
But from (2.9) and (2.17) we get
(4.6)g¯(Aξ*X,Z0)=-g¯(∇Xξ,Z0)=-g¯(∇¯Xξ,Z0)=g¯(ξ,∇¯XZ0)=g¯(ξ,∇XZ0+h(X,Z0))=0.
Therefore
(4.7)Aξ*X∉Γ(D0).
Also using (2.20), (2.21), and (2.24), we get
(4.8)g¯(Aξ*X,V)=-g¯(∇Xξ,V)=g¯(ξ,∇¯XV)=-g¯(ξ,ϕV)=g(V,ϕξ)=0.
Therefore
(4.9)Aξ*X∉{V}.
Hence from (4.4), (4.7), and (4.9) the result follows.Corollary 4.6.
LetM be a mixed geodesic GCR-lightlike submanifold of an indefinite Sasakian manifold M¯. Then g¯(hl(X,Y),ξ)=0, for any X∈Γ(D¯),Y∈Γ(M2) and ξ∈Γ(D2).Proof.
From (2.18) and above lemma, the result follows.Lemma 4.7.
LetM be a GCR-lightlike submanifold of an indefinite Sasakian manifold M¯. Then g¯(AWϕX,Y)=g¯(AWY,ϕX)-g¯(ϕX,Dl(Y,W)), for any X∈Γ(D⊕{V}),Y∈Γ(D¯) and W∈Γ(S(TM⊥)).Proof.
Using (2.12), we have
(4.10)g¯(AWϕX,Y)=-g¯(∇¯ϕXW,Y)=g¯(W,h(ϕX,Y))=g¯(∇¯YϕX,W)=-g¯(ϕX,∇¯YW)=g¯(ϕX,AWY)-g¯(ϕX,Dl(Y,W)),
for X∈Γ(D⊕{V}),Y∈Γ(D¯) and W∈Γ(S(TM⊥)).Theorem 4.8.
LetM be a mixed geodesic GCR-lightlike submanifold of an indefinite Sasakian manifold M¯. Then AUX∈Γ(D⊕{V}) and ∇XtU∈Γ(L⊥S), for any X∈Γ(D⊕{V}) and U∈Γ(L⊥S).Proof.
SinceM is mixed geodesic, therefore h(X,Y)=0 for any X∈Γ(D⊕{V}),Y∈Γ(D¯), therefore (2.7) gives
(4.11)0=∇¯XY-∇XY.
Since D¯ is anti-invariant there exist U∈Γ(L⊥S) such that ϕU=Y. Thus from (2.12), (3.6), and (3.7) we get
(4.12)0=∇¯XϕU-∇XY=(∇¯Xϕ)U+ϕ∇¯XU-∇XY=-g(X,U)V+ϵη(U)X+ϕ(-AUX+∇XtU)-∇XY=-g(X,U)V-fAUX-ωAUX+B∇XtU+C∇XtU-∇XY.
Comparing transversal components we get
(4.13)ωAUX=C∇XtU,
since ωAUX∈Γ(L⊥S) and C∇XtU∈Γ(L⊥S)⊥, hence AUX∈Γ(D⊕{V}) and ∇XtU∈Γ(L⊥S).


## 5. GCR-Lightlike Product

Definition 5.1.
AGCR-lightlike submanifold M of an indefinite Sasakian manifold M¯ is called GCR-lightlike product if both the distributions D⊕{V} and D¯ defines totally geodesic foliation in M.Theorem 5.2.
LetM be a GCR-lightlike submanifold of an indefinite Sasakian manifold M¯. Then the distribution D⊕{V} defines a totally geodesic foliation in M if and only if Bh(X,ϕY)=0, for any X,Y∈D⊕{V}.Proof.
SinceD¯=ϕ(L⊥S), therefore D⊕{V} defines a totally geodesic foliation in M if and only if
(5.1)g(∇XY,ϕξ)=g(∇XY,ϕW)=0,
for any X,Y∈Γ(D⊕{V}), ξ∈Γ(D2) and W∈Γ(S). Using (2.9) and (2.25), we have
(5.2)g(∇XY,ϕξ)=-g¯(∇¯XϕY,ξ)=-g¯(hl(X,fY),ξ),g(∇XY,ϕW)=-g¯(∇¯XϕY,W)=-g¯(hs(X,fY),W).
Hence, from (5.2) the assertion follows.Theorem 5.3.
LetM be a GCR-lightlike submanifold of an indefinite Sasakian manifold M¯. Then the distribution D¯ defines a totally geodesic foliation in M if and only if ANX has no component in ϕS⊥ϕD2 and AωYX has no component in D2⊥D0, for any X,Y∈Γ(D¯) and N∈Γ(ltr(TM)).Proof.
We know thatD¯ defines a totally geodesic foliation in M if and only if
(5.3)g(∇XY,N)=g(∇XY,ϕN1)=g(∇XY,V)=g(∇XY,ϕZ)=0,
for X,Y∈Γ(D¯), N∈Γ(ltr(TM)), Z∈Γ(D0), and N1∈Γ(L). Using (2.9) and (2.11) we have
(5.4)g(∇XY,N)=g¯(∇¯XY,N)=-g¯(Y,∇¯XN)=g(Y,ANX).
Using (2.9), (2.10), and (2.25) we obtain
(5.5)g(∇XY,ϕN1)=-g(ϕ∇¯XY,N1)=-g(∇¯XωY,N1)=g(AωYX,N1),g(∇XY,ϕZ)=-g(ϕ∇¯XY,Z)=-g(∇¯XωY,Z)=g(AωYX,Z),
also
(5.6)g(∇XY,V)=g(∇¯XY,V)=-g(Y,∇¯XV)=g(Y,ϕX)=0.
Thus from (5.4)–(5.6), the result follows.Theorem 5.4.
LetM be a GCR-lightlike submanifold of an indefinite Sasakian manifold M¯. If ∇f=0 then M is a GCR lightlike product.Proof.
LetX,Y∈Γ(D¯) therefore fY=0, then using (3.11) with the hypothesis, we get f∇XY=0 therefore the distribution D¯ defines a totally geodesic foliation.Let X,Y∈D⊕{V} therefore ωY=0 then using (3.9), we get
(5.7)Bh(X,Y)+g(X,Y)V+ϵη(Y)X=0.
Equating components along the distribution D¯ of above equation, we get Bh(X,Y)=0, therefore D⊕{V} define a totally geodesic foliation in M. Hence M is a GCR lightlike product.Definition 5.5.
A lightlike submanifoldM of a semi-Riemannian manifold is said to be an irrotational submanifold if ∇¯Xξ∈Γ(TM) for any X∈Γ(TM) and ξ∈ΓRad(TM). Thus M is an irrotational lightlike submanifold if and only if hl(X,ξ)=0,hs(X,ξ)=0.Theorem 5.6.
LetM be an irrotational GCR-lightlike submanifold of indefinite Sasakian manifold M¯. Then M is a GCR lightlike product if the following conditions are satisfied.  (A)
∇¯XU∈Γ(S(TM⊥),∀X∈Γ(TM),andU∈Γ(tr(TM)).(B)
Aξ*Y∈Γ(ϕ(S)),∀Y∈Γ(D).Proof.
Let(A) holds, then using (2.11) and (2.12), we get ANX=0,AWX=0,Dl(X,W)=0, and ∇XlN=0 for X∈Γ(TM). These equations imply that the distribution D¯ defines a totally geodesic foliation in M and with (2.13), we get g¯(hs(X,Y),W)=0. Hence non degeneracy of S(TM⊥) implies that hs(X,Y)=0. Therefore hs(X,Y) has no component in S. Finally, from (2.14) and M being irrotational, we have g¯(hl(X,Y),ξ)=g¯(Y,Aξ*X), for X∈Γ(TM) and Y∈Γ(D). Assume (B) holds, then hl(X,Y)=0. Therefore hl(X,Y) has no component in L. Thus the distribution D⊕{V} defines a totally geodesic foliation in M. Hence M is a GCR lightlike product.Definition 5.7 (see [6]).
If the second fundamental formh of a submanifold tangent to characteristic vector field V, of an indefinite Sasakian manifold M¯ is of the form
(5.8)h(X,Y)={g(X,Y)-η(X)η(Y)}α+η(X)h(Y,V)+η(Y)h(X,V),
for any X,Y∈Γ(TM), where α is a vector field transversal to M, then M is called a totally contact umbilical submanifold of an indefinite Sasakian manifold.Theorem 5.8.
LetM be a totally contact umbilical GCR-lightlike submanifold of an indefinite Sasakian manifold M¯. Then M is a GCR-lightlike product if Bh(X,Y)=0, for any X∈Γ(TM) and Y∈Γ(D⊕{V}).Proof.
LetX,Y∈Γ(D⊕{V}), then the hypothesis Bh(X,Y)=0, implies that the distribution D⊕{V} defines totally geodesic foliation in M.Let X,Y∈Γ(D¯) then using (3.9), we have
(5.9)-f∇XY=AωYX+Bh(X,Y)-g(X,Y)V,
let Z∈Γ(D0) and using (2.8) and (2.25), then above equation becomes
(5.10)-g(f∇XY,Z)=g(AωYX+Bh(X,Y)-g(X,Y)V,Z)=g(∇¯XϕY,Z)=-g(∇¯XY,ϕZ)=g(Y,∇XZ′),
where ϕZ=Z′∈Γ(D0). For X∈Γ(D¯) from (3.10), we get
(5.11)ωP∇XZ=h(X,fZ)-Ch(X,Z).
Using the hypothesis with (5.8), we get ωP∇XZ=0, this implies ∇XZ∈Γ(D) therefore, (5.10) becomes g(f∇XY,Z)=0. Then non degeneracy of the distribution D0 implies that the distribution D¯ defines a totally geodesic foliation in M. Hence the assertion follows.Theorem 5.9.
LetM be a totally geodesic GCR-lightlike submanifold of an indefinite Sasakian manifold M¯. Suppose there exists a transversal vector bundle of M which is parallel along D¯ with respect to Levi-Civita connection on M, that is, ∇¯XU∈Γ(tr(TM)), for any U∈Γ(tr(TM)), X∈Γ(D¯). Then M is a GCR-lightlike product.Proof.
SinceM is a totally geodesic GCR-lightlike, therefore Bh(X,Y)=0, for X,Y∈Γ(D⊕{V}), this implies D⊕{V} defines a totally geodesic foliation in M. Next, ∇¯XU∈Γ(tr(TM)) implies AUX=0. Hence by Theorem 5.3, the distribution D¯ defines a totally geodesic foliation in M. Thus M is a contact GCR-lightlike product.



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