# GCR-Lightlike Product of Indefinite Kaehler Manifolds

Authors: Rakesh Kumar; Sangeet Kumar; R. K. Nagaich
Journal: ISRN Geometry
 (2011)

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## Abstract

We study geodesicGCR-lightlike submanifolds of indefinite Kaehler 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 ofCR-submanifolds of Kaehler manifolds was initiated by Bejancu [1], which includes holomorphic and totally real submanifolds as subcases, and further developed by Bejancu [2], Bejancu et al. [3], Blair and Chen [4], Chen [5], Yano and Kon [6, 7], and many others. They all studied the geometry of CR-submanifolds with positive definite metric. Therefore this geometry may not be applicable to the other branches of mathematics and physics, where the metric is not necessarily definite. Thus the geometry of CR-submanifolds with indefinite metric became a topic of chief discussion and Duggal [8, 9] played a very crucial role. Duggal and Bejancu [10] introduced the notion of CR-lightlike submanifolds which exclude the totally real and complex subcases. Then Duggal and Sahin [11] introduced SCR-lightlike submanifolds which contain complex and totally real subcases but there was no inclusion relation between CR and SCR-cases. Thus to find a class of submanifolds which would behave as an umbrella for CR-lightlike and SCR-lightlike submanifolds of an indefinite Kaehler manifold, Duggal and Sahin [12] introduced GCR-lightlike submanifolds of indefinite Kaehler manifolds. This paper starts with a very brief introduction about lightlike geometry and GCR-lightlike submanifolds which will be needed throught the paper and then we study geodesic GCR-lightlike submanifolds and 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 [8] 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) is an m-dimensional submanifold of M¯ and g is 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.Screen distributionS(TM) 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}and {N1,…,Nr} are local lightlike bases of Γ(RadTM|u) and Γ(ltr(TM)|u), and {Wr+1,…,Wn} and {Xr+1,…,Xm} are local orthonormal bases of Γ(S(TM⊥)|u) and Γ(S(TM)|u), respectively. For this quasiorthonormal fields of frames, we have the following.Theorem 2.1 (see [8]).
Let(M,g,S(TM),S(TM⊥)) be an r-lightlike submanifold of a semi-Riemannian manifold (M¯,g¯). Then there exist 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 second fundamental form, and AU is a linear a operator on M and known as shape operator.According to (2.4) considering the projection morphisms L and S of tr(TM) on ltr(TM) and S(TM⊥), respectively, then (2.7) and (2.8) become(2.9)∇¯XY=∇XY+hl(X,Y)+hs(X,Y),(2.10)∇¯XU=-AUX+DXlU+DXsU,
where we put hl(X,Y)=L(h(X,Y)),hs(X,Y)=S(h(X,Y)),DXlU=L(∇X⊥U), and DXsU=S(∇X⊥U).Ashl and hs are Γ(ltr(TM))-valued and Γ(S(TM⊥))-valued, respectively, therefore they are called 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 (2.9)–(2.12) we obtain(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), we 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 are called as second fundamental forms of distributions S(TM) and RadTM, respectively.From the geometry of Riemannian submanifolds and nondegenerate submanifolds, it is known that the induced connection∇ on a nondegenerate submanifold is a metric connection. Unfortunately, this is not true for lightlike submanifolds. Indeed considering ∇¯ a metric connection, we have(2.18)(∇Xg)(Y,Z)=g¯(hl(X,Y),Z)+g¯(hl(X,Z),Y),
for any X,Y,Z∈Γ(TM). From [8, page 171], using the properties of linear connection, we have(2.19)(∇Xhl)(Y,Z)=∇Xl(hl(Y,Z))-hl(∇XY,Z)-hl(Y,∇XZ),(∇Xhs)(Y,Z)=∇Xs(hl(Y,Z))-hs(∇XY,Z)-hs(Y,∇XZ).
Barros and Romero [13] defined indefinite Kaehler manifolds as follows.Definition 2.2.
Let(M¯,J¯,g¯) be an indefinite almost Hermitian manifold and let ∇¯ be the Levi-Civita connection on M¯ with respect to g¯. Then M¯ is called an indefinite Kaehler manifold if J¯ is parallel with respect to ∇¯, that is,
(2.20)(∇¯XJ¯)Y=0,∀X,Y∈Γ(TM¯).


## 3. Generalized Cauchy-Riemann Lightlike Submanifolds

Definition 3.1.
Let(M,g,S(TM)) be a real lightlike submanifold of an indefinite Kaehler manifold (M¯,g¯,J¯), then M is called a 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,J¯(D1)=D1,J¯(D2)⊂S(TM).(B)
There exist two subbundlesD0 and D′ of S(TM) such that
(3.2)S(TM)={J¯D2⊕D′}⊥D0,J¯(D0)=D0,J¯(D′)=L1⊥L2, where D0 is a nondegenerate distribution on M, and L1 and L2 are vector bundle of ltr(TM) and S(TM)⊥, respectively.Then the tangent bundleTM of M is decomposed as(3.3)TM=D⊥D′,D=Rad(TM)⊕D0⊕J¯D2.M is called a proper GCR-lightlike submanifold if D1≠{0}, D2≠{0}, D0≠{0}, and L2≠{0}.LetQ, P1, and P2 be the projections on D, J¯(L1)=M1 and J¯(L2)=M2, respectively. Then for any X∈Γ(TM) we have(3.4)X=QX+P1X+P2X,
applying J¯ to (3.4), we obtain(3.5)J¯X=TX+wP1X+wP2X,
and we can write (3.5) as(3.6)J¯X=TX+wX,
where TX and wX are the tangential and transversal components of J¯X, respectively.Similarly(3.7)J¯V=BV+CV,
for any V∈Γ(tr(TM)), where BV and CV are the sections of TM and tr(TM), respectively.Differentiating (3.5) and using (2.9)–(2.12) and (3.7) we have(3.8)Ds(X,wP1Y)=-∇XswP2Y+wP2∇XY-hs(X,TY)+Chs(X,Y),Dl(X,wP2Y)=-∇XlwP1Y+wP1∇XY-hl(X,TY)+Chl(X,Y).
Using Kaehlerian property of ∇¯ with (2.11) and (2.12), we have the following lemmas.Lemma 3.2.
LetM be a GCR-lightlike submanifold of an indefinite Kaehlerian manifold M¯. Then one has
(3.9)(∇XT)Y=AwYX+Bh(X,Y),(3.10)(∇Xtw)Y=Ch(X,Y)-h(X,TY),
where X,Y∈Γ(TM) and
(3.11)(∇XT)Y=∇XTY-T∇XY,(3.12)(∇Xtw)Y=∇XtwY-w∇XY.Lemma 3.3.
LetM be a GCR-lightlike submanifold of an indefinite Kaehlerian manifold M¯. Then one has
(3.13)(∇XB)V=ACVX-TAVX,(∇XtC)V=-wAVX-h(X,BV),
where X∈Γ(TM), V∈Γ(tr(TM)), and
(3.14)(∇XB)V=∇XBV-B∇XtV,(∇XtC)V=∇XtCV-C∇XtV.Duggal and Sahin [12] investigated the conditions to define totally geodesic foliations by the distributions D and D′ in M as follows.Theorem 3.4 (see [12]).
LetM be a GCR-lightlike submanifold of an indefinite Kaehler manifold M¯. Then the distribution D defines a totally geodesic foliation in M if and only if Bh(X,Y)=0, for any X,Y∈Γ(D).Theorem 3.5 (see [12]).
LetM be a GCR-lightlike submanifold of an indefinite Kaehler manifold M¯. Then the distribution D′ defines a totally geodesic foliation in M if and only if AwYX∈Γ(D′), for any X,Y∈Γ(D′).


## 4. GeodesicGCR-Lightlike Submanifolds

Definition 4.1.
AGCR-lightlike submanifold of an indefinite Kaehler 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(D′).Definition 4.2.
AGCR-lightlike submanifold of an indefinite Kaehler manifold is called D geodesic GCR-lightlike submanifold if its second fundamental form h satisfies h(X,Y)=0 for any X,Y∈Γ(D).Definition 4.3.
AGCR-lightlike submanifold of an indefinite Kaehler manifold is called D′ geodesic GCR-lightlike submanifold if its second fundamental form h satisfies h(X,Y)=0 for any X,Y∈Γ(D′).Theorem 4.4.
LetM be a GCR-lightlike submanifold of an indefinite Kaehler manifold M¯. Then M is D-geodesic if and only if
(4.1)g(AWX,Y)=g¯(Dl(X,W),Y),∇X*J¯ξ∉Γ(D0⊥J¯L1),Aξ*Y∉Γ(J¯L1),hl(X,ξ)∉Γ(L1),
for any X,Y∈Γ(D),ξ∈Γ(Rad(TM)), and W∈Γ(S(TM⊥)).Proof.
Using the definition ofGCR-lightlike submanifolds, M is D-geodesic, if and only if
(4.2)g¯(hl(X,Y),ξ)=0,g¯(hs(X,Y),W)=0,
for any X,Y∈Γ(D),ξ∈Γ(Rad(TM)), and W∈Γ(S(TM⊥)). Thus for X,Y∈Γ(D), first part of the assertion follows from (2.13).
Now forX,Y∈Γ(D),ξ∈Γ(Rad(TM)) using (2.16), we have
(4.3)g¯(hl(X,Y),ξ)=g¯(∇¯XY,ξ)=-g¯(J¯Y,∇¯XJ¯ξ)=-g(J¯Y,∇XJ¯ξ)-g¯(J¯Y,hl(X,J¯ξ))=-g(J¯Y,∇X*J¯ξ)-g¯(J¯Y,hl(X,J¯ξ)).
Since Y∈Γ(D), this implies that Y∈Γ(D0), Y∈Γ(D1), Y∈Γ(D2), or Y∈Γ(J¯D2). If Y∈Γ(D0) or Y∈Γ(D2), then we have
(4.4)g¯(J¯Y,hl(X,J¯ξ))=0,
and if Y∈Γ(D1) or Y∈Γ(J¯D2), then we have
(4.5)g¯(J¯Y,hl(X,J¯ξ))=g(Aξ′*X,J¯ξ)+g¯(hl(X,ξ′),J¯ξ),
for any ξ′=J¯Y∈Γ(Rad(TM)).
Now using (4.4) and (4.5) in (4.3), we obtain
(4.6)g¯(hl(X,Y),ξ)=-g(J¯Y,∇X*J¯ξ)-g(Aξ′*X,J¯ξ)-g¯(hl(X,ξ′),J¯ξ).
Hence the second part of the assertion follows from (4.6).Theorem 4.5.
LetM be a GCR-lightlike submanifold of an indefinite Kaehler manifold M¯. Then M is D′-geodesic if and only if AWX and Aξ*X have no components in M2⊥J¯D2 for any X∈Γ(D′), ξ∈Γ(Rad(TM)), and W∈Γ(S(TM⊥)).Proof.
ForX,Y∈Γ(D′) and W∈Γ(S(TM⊥)) using (2.13), we obtain
(4.7)g¯(hs(X,Y),W)=g(AWX,Y),
and for ξ∈Γ(Rad(TM)) using (2.14) and (2.17) we obtain
(4.8)g¯(hl(X,Y),ξ)=g(Aξ*X,Y).
Hence the assertion follows from (4.7) and (4.8).Theorem 4.6.
LetM be a GCR-lightlike submanifold of an indefinite Kaehler manifold M¯. Then M is mixed geodesic if and only if
(4.9)Aξ*X∈Γ(D0⊥J¯L1),AWX∈Γ(D0⊥Rad(TM)⊥J¯L1),
for any X∈Γ(D),ξ∈Γ(Rad(TM)), and W∈Γ(S(TM⊥)).Proof.
For anyX∈Γ(D),Y∈Γ(D′), and ξ∈Γ(Rad(TM)) using (2.14) and (2.17) we obtain
(4.10)g¯(hl(X,Y),ξ)=g(Aξ*X,Y),
and for W∈Γ(S(TM⊥)) with (2.13), we obtain
(4.11)g¯(hs(X,Y),W)=g(AWX,Y).
Hence the result follows from (4.10) and (4.11).Theorem 4.7.
LetM be a mixed geodesic GCR-lightlike submanifold of an indefinite Kaehler manifold M¯. Then one has
(4.12)Aξ*X∈Γ(J¯D2),
for any X∈Γ(D′) and ξ∈Γ(D2).Proof.
ForX∈Γ(D′) and ξ∈Γ(D2) we have
(4.13)h(X,J¯ξ)=∇¯XJ¯ξ-∇XJ¯ξ=J¯∇¯Xξ-∇XJ¯ξ=J¯∇Xξ-J¯h(X,ξ)-∇XJ¯ξ.
Since M is mixed geodesic, therefore
(4.14)J¯∇Xξ=∇XJ¯ξ.
Using (2.16) and (2.17) we obtain
(4.15)-TAξ*X-wAξ*X+J¯∇X*tξ=∇X*J¯ξ+h*(X,J¯ξ).
Equating the transversal components we have
(4.16)wAξ*X=0.
Thus
(4.17)Aξ*X∈Γ(J¯D2⊥D0).
Now, for Z∈Γ(D0) and ξ∈Γ(D2) we have
(4.18)g¯(Aξ*X,Z)=g¯(-∇Xξ+∇X*tξ,Z)=-g¯(∇Xξ,Z)=-g¯(∇¯Xξ,Z)=g¯(ξ,∇¯XZ)=g¯(ξ,∇XZ+h(X,Z))=0.
If Aξ*X∈Γ(D0), then using the nondegeneracy of D0 for any Z∈Γ(D0), we have g¯(Aξ*X,Z)≠0. Therefore Aξ*X∉Γ(D0). Hence the assertion is proved.Theorem 4.8.
LetM be a mixed geodesic GCR-lightlike submanifold of an indefinite Kaehler manifold M¯. Then the transversal section V∈Γ(J¯D′) is D-parallel if and only if ∇XJ¯V∈Γ(D), for any X∈Γ(D).Proof.
LetY∈Γ(D′) such that J¯Y=wY=V∈Γ(L1⊥L2) and X∈Γ(D); then using hypothesis in (3.9) we have T∇XY=-AwYX=-AVX. Now ∇XtV=∇¯XV+AVX=∇¯XJ¯Y-T∇XY. Since ∇¯ is a Kaehlerian connection and M is mixed geodesic, therefore we have ∇XtV=w∇XY or consequently ∇XtV=-w∇XJ¯V, which clearly proves the theorem.Theorem 4.9.
LetM be a GCR-lightlike submanifold of an indefinite Kaehler manifold M¯ such that Ds(X,V)∈Γ(L2⊥). Then AJ¯VX=J¯AVX for any X∈Γ(D) and V∈Γ(L1⊥).Proof.
ForX∈Γ(D),Y∈Γ(D′), and V∈Γ(L1⊥), we have
(4.19)g(AJ¯VX-J¯AVX,Y)=g(AJ¯VX,Y)-g(J¯AVX,Y)=g(AJ¯VX,Y)+g(AVX,J¯Y)=-g(∇¯XJ¯V,Y)-g(∇¯XV,J¯Y)=-g(J¯∇¯XV,Y)+g(J¯∇¯XV,Y)=0.
For X∈Γ(D),Z∈Γ(D0), and V∈Γ(L1⊥), we have
(4.20)g(AJ¯VX-J¯AVX,Z)=g(AJ¯VX,Z)-g(J¯AVX,Z)=g(AJ¯VX,Z)+g(AVX,J¯Z)=-g(∇¯XJ¯V,Z)-g(∇¯XV,J¯Z)=-g(J¯∇¯XV,Z)+g(J¯∇¯XV,Z)=0.
For X∈Γ(D),N∈Γ(ltr(TM)), and V∈Γ(L1⊥), we have
(4.21)g(AJ¯VX-J¯AVX,N)=g(AJ¯VX,N)-g(J¯AVX,N)=g(AJ¯VX,N)+g(AVX,J¯N)=-g(∇¯XJ¯V,N)-g(∇¯XV,J¯N)=-g(J¯∇¯XV,N)+g(J¯∇¯XV,N)=0.
For X∈Γ(D),J¯N∈Γ(J¯L1), and V∈Γ(L1⊥), we have
(4.22)g(AJ¯VX-J¯AVX,J¯N)=g(AJ¯VX,J¯N)-g(J¯AVX,J¯N)=g(AJ¯VX,J¯N)-g(AVX,N)=-g(∇¯XJ¯V,J¯N)+g(∇¯XV,N)=-g(J¯∇¯XV,J¯N)+g(∇¯XV,N)=-g(∇¯XV,N)+g(∇¯XV,N)=0.
Hence the assertion follows from (4.19)–(4.22).


## 5.GCR-Lightlike Product

Definition 5.1.
AGCR-lightlike submanifold M of an indefinite Kaehler manifold M¯ is called a GCR-lightlike product if both the distributions D and D′ define totally geodesic foliations in M.Lemma 5.2.
LetM be a totally umbilical GCR-lightlike submanifold of an indefinite Kaehler manifold M¯; then the distribution D′ defines a totally geodesic foliation in M.Proof.
For anyX,Y∈Γ(D′), (3.9) implies that T∇XY=-AwYX-Bh(X,Y); then for Z∈Γ(D0) we have
(5.1)g(T∇XY,Z)=-g(AwYX,Z)-g(Bh(X,Y),Z)=g(∇¯XwY,Z)=g(∇¯XJ¯Y,Z)=-g(∇¯XY,J¯Z)=-g(∇¯XY,Z′)=g(Y,∇XZ′),
where Z′=J¯Z∈Γ(D0). Since X∈Γ(D′) and Z∈Γ(D0), then from (3.8) we have wP∇XZ=h(X,TZ)-Ch(X,Z)=Hg(X,TZ)-CHg(X,Z)=0, therefore wP∇XZ=0, and this implies that ∇XZ∈Γ(D). Therefore (5.1) implies that g(T∇XY,Z)=0; then the nondegeneracy of D0 implies that T∇XY=0. Hence ∇XY∈Γ(D′), for any X,Y∈Γ(D′). Thus the result follows.Theorem 5.3.
LetM be a totally umbilical GCR-lightlike submanifold of an indefinite Kaehler manifold M¯. Then M is a GCR-lightlike product if and only if Bh(X,Y)=0, for any X∈Γ(TM) and Y∈Γ(D).Proof.
LetM be a GCR-lightlike product; therefore the distributions D and D′ define a totally geodesic foliation in M. Therefore using Theorem 3.4, Bh(X,Y)=0 for any X,Y∈Γ(D). Now let X∈Γ(D′) and Y∈Γ(D); then Bh(X,Y)=g(X,Y)BH=0. Hence Bh(X,Y)=0, for any X∈Γ(TM) and Y∈Γ(D).
Conversely, letX,Y∈Γ(D); then Bh(X,Y)=0 implies that D defines a totally geodesic foliation in M. Let X,Y∈Γ(D′); then (3.9) and (3.11) imply that AwYX=-T∇XY-Bh(X,Y). Using Lemma 5.2, we obtain TAwYX+wAwYX=-h(X,Y), we compare the tangential components, we get TAwYX=0, and this implies that AwYX∈Γ(D′). Hence using Theorem 3.5, the distribution D′ defines a totally geodesic foliation in M. Consequently, M is a GCR-lightlike product of an indefinite Kaehler manifold.Theorem 5.4.
LetM be a totally geodesic GCR-lightlike submanifold of an indefinite Kaehler manifold M¯. Suppose that there exists a transversal vector bundle of M, which is parallel along D′ with respect to the Levi-Civita connection on M; that is, one has ∇¯XV∈Γ(tr(TM)), for any V∈Γ(tr(TM)) and X∈Γ(D′). Then M is a GCR-lightlike product.Proof.
SinceM is a totally geodesic GCR-lightlike submanifold, therefore Bh(X,Y)=0, for any X,Y∈Γ(D). Hence the distribution D defines a totally geodesic foliation in M. Next, since ∇¯XV∈Γ(tr(TM)) for any V∈Γ(tr(TM)) and X∈Γ(D′), therefore using (2.8), we have AVX=0 then using (3.9), we obtain T∇XY=0 for any X,Y∈Γ(D′) and this implies that ∇XY∈Γ(D′). Hence the distribution D′ defines a totally geodesic foliation in M. Thus 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 lightlike submanifold of an indefinite Kaehler manifold M¯. Then M is a GCR-lightlike product if the following conditions are satisfied:  (A)
∇¯XU∈Γ(S(TM⊥)for allX∈Γ(TM)andU∈Γtr(TM),(B)
Aξ*Y∈Γ(J¯L2)for allY∈Γ(D).Proof.
Using (2.11) and (2.12) with (A), we get AWX=0,Dl(X,W)=0, and ∇XlW=0 for any X∈Γ(TM) and W∈Γ(S(TM⊥)). Therefore using (2.13) we have g(hs(X,Y),W)=0; then nondegeneracy of S(TM⊥) implies that hs(X,Y)=0. Hence Bhs(X,Y)=0. Now, let X,Y∈Γ(D) and ξ∈Γ(Rad(TM)); then using (B), we have g(hl(X,Y),ξ)=-g(∇Xξ,Y)=g(Aξ*X,Y)=0. Then using (2.6), we get hl(X,Y)=0. Hence Bhl(X,Y)=0. Thus the distribution D defines a totally geodesic foliation in M.
Next, letX,Y∈Γ(D′); then JY=wY∈Γ(L1⊥L2)⊂tr(TM). Using (3.9) we obtain T∇XY=-Bh(X,Y), comparing the components along D we get T∇XY=0, and this implies that ∇XY∈Γ(D′). Thus the distribution D′ defines a totally geodesic foliation in M. Hence M is a GCR-lightlike product.Theorem 5.7.
LetM be a GCR-lightlike submanifold of an indefinite Kaehler manifold M¯. Then M is a GCR-lightlike product if and only if (∇XT)Y=0, for any X,Y∈Γ(D) or X,Y∈Γ(D′).Proof.
Let(∇XT)Y=0, for any X,Y∈Γ(D) or X,Y∈Γ(D′). Let X,Y∈Γ(D), then wY=0 and (3.9) gives that Bh(X,Y)=0. Hence using Theorem 3.4, the distribution D defines a totally geodesic foliation in M. Next, let X,Y∈Γ(D′). Since BV∈Γ(D′) for any V∈Γ(tr(TM)), then (3.9) implies that AwYX∈Γ(D′). Hence using Theorem 3.5, the distribution D′ defines a totally geodesic foliation in M. Since both the distributions D and D′ define totally geodesic foliations in M, hence M is a GCR-lightlike product.
Conversely, letM be a GCR-lightlike product; therefore the distribution D defines a totally geodesic foliation in M. Using Kaehlerian property of ∇¯, for any X,Y∈Γ(D) we have ∇¯XJ¯Y=J¯∇¯XY; then comparing transversal components, we obtain h(X,J¯Y)=J¯h(X,Y) and then (∇XT)Y=∇XTY-T∇XY=∇¯XJ¯Y-h(X,J¯Y)-J¯∇¯XY+h(X,J¯Y)=0, that is, (∇XT)Y=0, for any X,Y∈Γ(D). Let D′ defines a totally geodesic foliation in M, and using Kaehlerian property of ∇¯, we have ∇¯XJ¯Y=J¯∇¯XY; then comparing tangential components on both sides, we obtain -AwYX=Bh(X,Y); then (3.9) implies that (∇XT)Y=0, which completes the proof.



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*Source: 531281-2011-08-07.xml*