We have studied mixed foliate CR-lightlike submanifolds and
CR-lightlike product of an indefinite Kaehler manifold and also obtained relationship
between them. Mixed foliate CR-lightlike submanifold of indefinite
complex space form has also been discussed and showed that the indefinite
Kaehler manifold becomes the complex semi-Euclidean space.

1. Introduction

The geometry of CR-submanifolds of Kaehler manifolds was initiated by Bejancu [1] and has been developed by [2–5] and others. They studied the geometry of CR-submanifolds with positive definite metric. Thus, this geometry may not be applicable to the other branches of mathematics and physics, where the metric is not necessarily definite. Moreover, because of growing importance of lightlike submanifolds and hypersurfaces in mathematical physics, especially in relativity, make the geometry of lightlike submanifolds and hypersurfaces a topic of chief discussion in the present scenario. In the establishment of the general theory of lightlike submanifolds and hypersurfaces, Kupeli [6] and Duggal and Bejancu [7] played a very crucial role. The objective of this paper is to study CR-lightlike submanifolds of an indefinite Kaehler manifold. This motivated us to study CR-lightlike submanifolds extensively.

2. Lightlike Submanifolds

We recall notations and fundamental equations for lightlike submanifolds, which are due to [7] 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, (M,g) an m-dimensional submanifold of M¯, and g 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,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 mappingRadTM:x∈M⟶RadTxM
defines a smooth distribution on M of rank r>0, then the submanifold M of M¯ is called r-lightlike submanifold and RadTM is called the radical distribution on M.

Let S(TM) be a screen distribution which is a semi-Riemannian complementary distribution of Rad(TM) in TM, that is,TM=RadTM⊥S(TM),
where 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 havetr(TM)=ltr(TM)⊥S(TM⊥),TM¯∣M=TM⊕tr(TM)=(RadTM⊕ltr(TM))⊥S(TM)⊥S(TM⊥).
Let u 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 [<xref ref-type="bibr" rid="B9">7</xref>]).

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
g¯(Ni,ξj)=δij,g¯(Ni,Nj)=0,
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∇¯XY=∇XY+h(X,Y),∀X,Y∈Γ(TM),∇¯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 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, (2.7) and (2.8) give∇¯XY=∇XY+hl(X,Y)+hs(X,Y),∇¯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).

As hl 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,∇¯XN=-ANX+∇XlN+Ds(X,N),∇¯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 (2.9)–(2.12), we obtaing¯(hs(X,Y),W)+g¯(Y,Dl(X,W))=g(AWX,Y),g¯(hl(X,Y),ξ)+g¯(Y,hl(X,ξ))+g(Y,∇Xξ)=0,g¯(ANX,N′)+g¯(N,AN′X)=0,
for any ξ∈Γ(RadTM), W∈Γ(S(TM⊥)), and N,N′∈Γ(ltr(TM)).

Let P¯ is a projection of TM on S(TM). Now, considering the decomposition (2.3), we can write∇XP¯Y=∇X*P¯Y+h*(X,P¯Y),∇Xξ=-Aξ*X+∇X*tξ,
for any X,Y∈Γ(TM) and ξ∈Γ(RadTM), where {∇X*P¯Y,Aξ*X} and {h*(X,P¯Y),∇X*tξ} belong to Γ(S(TM)) and Γ(RadTM), respectively. Here, ∇* and ∇X*t are linear connections on S(TM) and RadTM, respectively. By using (2.9)-(2.10) and (2.14), we obtain
g¯(hl(X,P¯Y),ξ)=g(Aξ*X,P¯Y),g¯(h*(X,P¯Y),N)=g¯(ANX,P¯Y).

Definition 2.2.

Let (M¯,J¯,g¯) be a real 2m-dimensional indefinite Kaehler manifold and M an n-dimensional submanifold of M¯. Then, M is said to be a CR-lightlike submanifold if the following two conditions are fulfilled:

J¯(RadTM) is distribution on M such that
RadTM∩J¯(RadTM)=0,

there exist vector bundles S(TM),S(TM⊥),ltr(TM),D0 and D′ over M such that

S(TM)={J¯(RadTM)⊕D′}⊥D0,J¯(D0)=D0,J¯(D′)=L1⊥L2,
where Γ(D0) is a nondegenerate distribution on M, Γ(L1) and Γ(L2) are vector sub-bundles of Γ(ltr(TM)) and Γ(S(TM⊥)), respectively.

Clearly, the tangent bundle of a CR-lightlike submanifold is decomposed asTM=D⊕D′,
where
D=RadTM⊥J¯(RadTM)⊥D0.

Theorem 2.3.

Let M be a 1-lightlike submanifold of codimension 2 of a real 2 m-dimensional indefinite almost Hermitian manifold (M¯,J¯,g¯) such that J¯(RadTM) is a distribution on M. Then, M is a CR-lightlike submanifold.

Proof.

Since g¯(J¯ξ,ξ)=0, therefore, J¯ξ is tangent to M. Moreover, RadTM and J¯(RadTM) are distributions of rank 1 on M, therefore, RadTM∩J¯(RadTM)={0}. This enables one to choose a screen distribution S(TM) such that it contains J¯(RadTM). For any W∈Γ(S(TM⊥)), we have g¯(J¯W,ξ)=-g¯(W,J¯ξ)=0 and g¯(J¯W,W)=0. As S(TM⊥) is of rank 1, therefore, J¯(S(TM⊥)) is also a distribution on M such that J¯(S(TM⊥))∩{TM⊥⊥J¯(TM⊥)}={0} and S(TM)={J¯(RadTM)⊥J¯(S(TM⊥))}⊕E, where E is a distribution of rank 2 m–5. Now, for any N∈Γ(ltr(TM)), we have g¯(J¯N,ξ)=-g¯(N,J¯ξ)=0 and g¯(J¯N,W)=0, therefore, J¯N is tangent to M and g¯(J¯N,N)=g¯(J¯N,J¯N)=g¯(J¯N,J¯W)=0. This implies that J¯N has no component in RadTM, J¯(RadTM), and J¯(S(TM⊥)). Thus, J¯N lies in Γ(E) and we have
S(TM)={J¯(RadTM)⊕J¯(ltr(TM))}⊥J¯(S(TM⊥))⊥D0,
where D0 is a nondegenerate distribution; otherwise S(TM) would be degenerate.

Now, denote D′=J¯(ltr(TM))⊥J¯(S(TM⊥)), then condition (B) is satisfied, where J¯D′=L1⊥L2, L1=ltrTM, and L2=S(TM⊥). Hence the result is present.

3. Mixed Geodesic and <italic>CR</italic>-Lightlike ProductDefinition 3.1.

A CR-lightlike submanifold of an indefinite almost Hermitian manifold is called mixed geodesic CR-lightlike submanifolds if the second fundamental form h satisfies h(X,Y)=0, for any X∈Γ(D) and Y∈Γ(D′).

Definition 3.2.

A CR-lightlike submanifold of an indefinite almost Hermitian manifold is called D-geodesic (resp., D′-geodesic) CR-lightlike submanifolds if its second fundamental form h, satisfies h(X,Y)=0, for any X,Y∈Γ(D) (resp., X,Y∈Γ(D′)).

Definition 3.3 (see [<xref ref-type="bibr" rid="B11">8</xref>]).

A CR-lightlike submanifold M in a Kaehler manifold is said to be a mixed foliate if the distribution D is integrable and M is mixed totally geodesic CR-lightlike submanifold.

Now, suppose {N1,N2,…,Nr} be a basis of Γ(ltr(TM)) with respect to the basis {ξ1,ξ2,…,ξr} of Γ(RadTM), such that {N1,N2,…,Np} is a basis of Γ(L1). Also, consider an orthonormal basis {W1,W2,…,Ws} of Γ(L2).

Theorem 3.4.

Let M be a CR -lightlike submanifold of an indefinite Kaehler manifold M¯. Then, the distribution D defines a totally geodesic foliation if M is D-geodesic.

Proof.

Distribution D defines a totally geodesic foliation, if and only if,
∇XY∈Γ(D),∀X,Y∈Γ(D).

Since D′=J¯(L1⊥L2), therefore (3.1) holds, if and only if, g¯(∇XY,J¯ξi)=0, g¯(∇XY,J¯Wα)=0, for all i∈{1,…,p},α∈{1,…,s}.

Now let X,Y∈Γ(D), then g¯(∇XY,J¯ξi)=g¯(∇¯XY,J¯ξi)=-g¯(h(X,J¯Y),ξi), similarly g¯(∇XY,J¯Wα)=-g¯(h(X,J¯Y),Wα). Then, the result follows directly by using hypothesis.

Corollary 3.5.

Let M be a CR -lightlike submanifold of an indefinite Kaehler manifold M¯. If the distribution D defines a totally geodesic foliation, then hs(X,J¯Y)=0, for all X,Y∈Γ(D).

Theorem 3.6 (See [<xref ref-type="bibr" rid="B9">7</xref>]).

Let M be a CR -lightlike submanifold of an indefinite Kaehler manifold M¯. Then, the almost complex distribution D is integrable, if and only if, the second fundamental form of M satisfies
h(X,J¯Y)=h(J¯X,Y),
for any X,Y∈Γ(D).

Now, let S and Q be the projections on D and D′, respectively. Then, one hasJ¯X=fX+wX,∀X∈Γ(TM),
where fX=J¯SX and wX=J¯QX. Clearly, f is a tensor field of type (1,1) and w is Γ(L1⊥L2)-valued 1-form on M. Clearly, X∈Γ(D) if and only if, wX=0. On the other hand, one sets
J¯V=BV+CV,∀V∈Γ(tr(TM)),
where BV and CV are sections of TM and tr(TM), respectively.

By using Kaehlerian property of ∇¯ with (2.7) and (2.8), one has the following lemmas.

Lemma 3.7.

Let M be a CR -lightlike submanifold of an indefinite Kaehler manifold M¯. Then, one has
(∇Xf)Y=AwYX+Bh(X,Y),(∇Xtw)Y=Ch(X,Y)-h(X,fY),
for any X,Y∈Γ(TM), where
(∇Xf)Y=∇XfY-f(∇XY),(∇Xtw)Y=∇XtwY-w(∇XY).

Lemma 3.8.

Let M be a CR -lightlike submanifold of an indefinite Kaehler manifold M¯. Then, one has
(∇XB)V=-fAVX+ACVX,(∇XC)V=-wAVX-h(X,BV),
for any X∈Γ(TM) and V∈Γ(tr(TM)), where
(∇XB)V=∇XBV-B∇XtV,(∇XC)V=∇XtCV-C∇XtV.

Theorem 3.9.

Let M be a CR -lightlike submanifold of an indefinite Kaehler manifold M¯. Then, distribution D defines totally geodesic foliation, if and only if, D is integrable.

Proof.

From (3.6), we obtain
h(X,J¯Y)=Ch(X,Y)+w(∇XY),
for all X,Y∈Γ(D). Then, taking into account that h is symmetric and ∇ is torsion free, we obtain
h(X,J¯Y)-h(Y,J¯X)=w(∇XY)-w(∇YX),
which proves the assertion.

Corollary 3.10.

If CR -lightlike submanifold M in an indefinite Kaehler manifold is mixed foliate, then, using Theorem 3.9 and Corollary 3.5, it is clear that h(X,Y)=0, for any X∈Γ(D),Y∈Γ(D′), and hs(X,J¯Y)=0, for any X,Y∈Γ(D).

Theorem 3.11.

Let M be a CR -lightlike submanifold of an indefinite Kaehler manifold M¯. Then, M is mixed geodesic, if and only if, AwZX∈Γ(D), ∇XtwZ∈Γ(L1⊥L2), for any X∈Γ(D), Z∈Γ(D′).

Proof.

Since M¯ is a Kaehler manifold, therefore h(X,Y)=-J̅∇̅XJ¯Z-∇XZ=-J¯{-AJ¯ZX+∇XtJ¯Z}-∇XZ. Since Z∈Γ(D′), this implies that J¯Z=wZ, fZ=0, therefore, h(X,Y)=J¯(AJ¯ZX)-J¯(∇XtJ¯Z)-∇XZ. This gives h(X,Y)=fAwZX+wAwZX-B∇XtwZ-C∇XtwZ-∇XZ; comparing transversal parts, we obtain h(X,Y)=wAwZX-C∇XtwZ. Thus, M is mixed geodesic, if and only if, wAwZX=0,C∇XtwZ=0. This implies AwZX∈Γ(D),∇XtwZ∈Γ(L1⊥L2).

Lemma 3.12.

Let M be a CR -lightlike submanifold of an indefinite Kaehler manifold M¯. Then,
∇XZ=-fAWX+B∇XsW+BDl(X,W),
for any X∈Γ(TM),Z∈Γ(J¯L2) and W∈Γ(L2).

Proof.

Let W∈Γ(L2) such that Z=J¯W. Since M¯ is a Kaehler manifold, therefore, ∇¯XZ=J̅∇̅XW, for any X∈Γ(TM). Then, ∇XZ+h(X,Z)=J¯(-AWX+∇XsW+Dl(X,W)=-fAWX-wAWX+B∇XsW+C∇XsW+BDl(X,W)+CDl(X,W). Comparing the tangential parts, we obtain (3.9).

Let M¯ be an indefinite Kaehler manifold of constant holomorphic sectional curvature c, then curvature tensor is given byR¯(X,Y)Z=c4{g¯(Y,Z)X-g¯(X,Z)Y+g¯(J¯Y,Z)J¯X-g¯(J¯X,Z)J¯Y+2g¯(X,J¯Y)J¯Z}.

Theorem 3.13.

Let M be a mixed foliate CR -lightlike submanifold of an indefinite complex space form M¯(c). Then, M¯ is a complex semi-Euclidean space.

Proof.

From (3.16), we have
g¯(R¯(X,J¯X)Z,J¯Z)=-c2g(X,X)g(Z,Z),
for any X∈Γ(D0) and Z∈Γ(J¯L2). Since, M is a mixed foliate, therefore, for X∈Γ(D0) and Z∈Γ(J¯L2), we have
g¯(R¯(X,J¯X)Z,J¯Z)=g¯((∇Xhs)(J¯X,Z)-(∇J¯Xhs)(X,Z),J¯Z),
where (∇Xhs)(J¯X,Z)=∇Xs(hs(J¯X,Z))-hs(∇XJ¯X,Z)-hs(J¯X,∇XZ) and (∇J¯Xhs)(X,Z)=∇J¯Xs(hs(X,Z))-hs(∇J¯XX,Z)-hs(X,∇J¯XZ). By hypothesis, M is mixed geodesic, therefore, (∇Xhs)(J¯X,Z)-(∇J¯Xhs)(X,Z)=hs(∇J¯XX,Z)+hs(X,∇J¯XZ)-hs(∇XJ¯X,Z)-hs(J¯X,∇XZ). Again, by using hypothesis and Theorem 3.9, the distribution D defines totally geodesic foliation. Hence, ∇J¯XX,∇XJ¯X∈Γ(D). Therefore, we have (∇Xhs)(J¯X,Z)-(∇J¯Xhs)(X,Z)=hs(X,∇J¯XZ)-hs(J¯X,∇XZ); using Lemma 3.12 here, we obtain (∇Xhs)(J¯X,Z)-(∇J¯Xhs)(X,Z)=-hs(X,fAWJ¯X)+hs(X,B∇J¯XsW)+hs(X,BDl(J¯X,W))+hs(J¯X,fAWX)-hs(J¯X,B∇XsW)-hs(J¯X,BDl(X,W)).

By using, M is mixed geodesic and Corollary 3.10, we obtain(∇Xhs)(J¯X,Z)-(∇J¯Xhs)(X,Z)=0.
Therefore, from (3.17)–(3.19), we get (c/2)g(X,X)g(Z,Z)=0. Since D0 and J¯L2 are non-degenerate, therefore, we have c=0, which completes the proof.

Definition 3.14 (see [<xref ref-type="bibr" rid="B9">7</xref>]).

A CR-lightlike submanifold M of a Kaehler manifold M¯ is called a CR-lightlike product if both the distribution D and D′ define totally geodesic foliations on M.

Theorem 3.15 (see [<xref ref-type="bibr" rid="B9">7</xref>]).

Let M be a CR -lightlike submanifold of an indefinite Kaehler manifold M¯. Then, D defines a totally geodesic foliation on M, if and only if, for any X,Y∈Γ(D), h(X,Y) has no component in Γ(L1⊥L2).

Theorem 3.16 (see [<xref ref-type="bibr" rid="B9">7</xref>]).

Let M be a CR -lightlike submanifold of an indefinite Kaehler manifold M¯. Then, D′ defines a totally geodesic foliation on M, if and only if, for any X,Y∈Γ(D′), one has
h*(X,Y)=0,AJ¯YXhasnocomponentinD0,AJ¯YX+Bh(X,Y)hasnocomponentinRadTM.

Theorem 3.17 (see [<xref ref-type="bibr" rid="B4">3</xref>]).

A CR -submanifold of a Kaehler manifold M¯ is a CR -product, if and only if, P is parallel, that is, ∇¯P=0, where J¯X=PX+FX.

Now, we will give the characterization of CR-lightlike product in the following form.

Theorem 3.18.

A CR -lightlike submanifold of an indefinite Kaehler manifold M¯ is a CR -lightlike product, if and only if, f is parallel, that is, ∇f=0.

Proof.

If f is parallel, then (3.5), gives
Bh(X,Y)=-AwYX,
for any X,Y∈Γ(TM). In particular, if X∈Γ(D), then wX=0. Hence, (3.23) implies Bh(X,Y)=0, thus, Theorem 3.15 follows.

Let X,Y∈Γ(D′), then J¯Y=wY, then (3.23) implies AJ¯YX+Bh(X,Y)=0, and, hence, (3.22) follows. For any Z∈Γ(D0), g¯(AJ¯YX,Z)=g¯(AwYX,Z)=-g¯(Bh(X,Y),Z)=g¯(h(X,Y),J¯Z)=0. Thus, (3.21) holds well. Since (∇Xf)Y=0 for any X,Y∈Γ(TM), let X,Y∈Γ(D′), then we have f∇XY=0. Hence, ∇XY∈Γ(D′). Therefore, g¯(∇XY,N)=0, for any X,Y∈Γ(D′) and N∈Γ(ltr(TM)) and consequently h*(X,Y)=0. Hence, (3.20) holds well. Thus, M is a CR-lightlike product in M¯.

Conversely, if M is a CR-lightlike product, then by definition, D and D′ both defines totally geodesic foliation, that is, ∇XY∈Γ(D), for any X,Y∈Γ(D) and ∇XY∈Γ(D′), for any X,Y∈Γ(D′). If ∇XY∈Γ(D) for any X,Y∈Γ(D), then by comparing the transversal parts in the simplification of Kaehlerian property of ∇¯, we get
h(X,J¯Y)=J¯h(X,Y).
Therefore, from (2.7), (3.7), and (3.24), we may prove that (∇Xf)Y=0 or ∇f=0.

Let ∇XY∈Γ(D′), for any X,Y∈Γ(D′), then by comparing the tangential parts in the simplification of Kaehlerian property of ∇¯, we getBh(X,Y)+AwYX=0.
Thus, from (3.5) and (3.25), we obtain (∇Xf)Y=0 or ∇f=0. Hence, the proof is complete.

Theorem 3.19.

Let M be a CR -lightlike submanifold of an indefinite Kaehler manifold M¯. If M is a CR -lightlike product, then it is a mixed foliate CR-lightlike submanifold.

Proof.

Since M is a CR-lightlike product then by Theorem 3.9, it is clear that the distribution D is integrable. Also, by using Theorem 3.18, for CR-lightlike product, ∇f=0, then (3.5) gives
Bh(X,Y)=-AwYX,
for any X,Y∈Γ(TM). Then, in particular, if X∈Γ(D), then wX=0. Hence, (3.26) implies Bh(X,Y)=0, that is,
AJ¯ZX=0,
for any Z∈Γ(D′) and X∈Γ(D).

Next, by the definition of CR-lightlike submanifold M is mixed geodesic, if and only if, g¯(h(X,Y),ξ)=0 and g¯(h(X,Y),W)=0, for any X∈Γ(D), Y∈Γ(D′), and W∈Γ(S(TM⊥)). By hypothesis, we have (∇Xf)Y=0 for any X,Y∈Γ(TM); let X∈Γ(D) and Y∈Γ(D′), then we have f∇XY=0. Hence, ∇XY∈Γ(D′). Therefore, g¯(h(X,Y),ξ)=g¯(∇¯XY,ξ), since Y∈Γ(D′), therefore, there exists V∈Γ(L1⊥L2) such that Y=J¯V; this gives g¯(h(X,Y),ξ)=g¯(∇¯XJ¯V,ξ)=-g¯(∇¯XV,J¯ξ)=g¯(AVX,J¯ξ)=-g¯(AJ¯YX,J¯ξ)=0, by virtue of (3.27).

Similarly, g¯(h(X,Y),W)=g¯(J¯∇̅XV,W)=-g¯(C∇XtV,W). Now, from (3.10) and (3.12), we have C∇XtV=∇XtCV+wAVX+h(X,BV); since V∈Γ(L1⊥L2), therefore CV=0, so we have C∇XtV=wAVX+h(X,BV). Therefore, by using (3.27), we have g¯(h(X,Y),W)=-g¯(h(X,BV),W)=-g¯(h(X,J¯Y),W); this gives g¯(h(X,Y),W)=0, for all X∈Γ(D),Y∈Γ(D′). Hence, the proof is complete.

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