# Characterization of Holomorphic Bisectional Curvature ofGCR-Lightlike Submanifolds

Authors: Sangeet Kumar; Rakesh Kumar; R. K. Nagaich
Journal: Advances in Mathematical Physics
 (2012)

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## Abstract

We obtain the expressions for sectional curvature, holomorphic sectional curvature, and holomorphic bisectional curvature of aGCR-lightlike submanifold of an indefinite Kaehler manifold. We discuss the boundedness of holomorphic sectional curvature of GCR-lightlike submanifolds of an indefinite complex space form. We establish a condition for a GCR-lightlike submanifold of an indefinite complex space form to be null holomorphically flat. We also obtain some characterization theorems for holomorphic sectional and holomorphic bisectional curvature.


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## Body

## 1. Introduction

The study ofCR submanifolds of Kaehler manifolds was initiated by Bejancu [1], as a generalization of totally real and complex submanifolds and further developed by [2–7]. The CR structures on real hypresurfaces of complex manifolds have interesting applications to relativity. Penrose [8] discovered a correspondence, called Penrose correspondence, between points of a Minkowski space and projective lines of a certain real hypersurfaces in a complex projective space, which is an interesting means of passing from the geometry of a Minkowski space to the geometry of a CR manifold. Duggal [9, 10] studied the geometry of CR submanifolds with Lorentzian metric and obtained their interaction with relativity. The theory of lightlike submanifolds has interaction with some results on Killing horizon, electromagnetic, and radition fields and asymptotically flat spacetimes (for detail see chapters 7, 8, and 9 of [11]). Thus due to the significant applications of CR structures in relativity and growing importance of lightlike submanifolds in mathematical physics and relativity, Duggal and Bejancu [11] introduced the notion of CR-lightlike submanifolds of indefinite Kaehler manifolds which have direct relation with physically important asymptotically flat space time which further lead to Twistor theory of Penrose and Heaven theory of Newman. Moreover, they concluded that, contrary to the CR-non degenerate submanifolds, CR-lightlike submanifolds do not include invariant (complex) and totally real lightlike submanifolds. Therefore, Duggal and Sahin [12] introduced SCR-lightlike submanifolds of indefinite Kaehler manifold which contain complex and totally real subcases but there was no inclusion relation between CR and SCR cases. Later on, Duggal and Sahin [13] introduced GCR-lightlike submanifolds of indefinite Kaehler manifolds, which behaves as an umbrella of invariant (complex), screen real and CR-lightlike submanifolds and also studied the existence (or nonexistence) of this new class in an indefinite space form. R. Kumar et al. [14] studied geodesic GCR-lightlike submanifolds of indefinite Kaehler manifolds and obtained some characterization theorems for a GCR-lightlike submanifold to be a GCR-lightlike product.Since sectional curvature offers a lot of information concerning the intrinsic geometry of Riemannian manifolds, therefore in this paper, we obtain the expressions for sectional curvature, holomorphic sectional curvature, and holomorphic bisectional curvature of aGCR-lightlike submanifold of an indefinite Kaehler manifold. In [15], Kulkarni showed that the boundedness of the sectional curvature on a semi-Riemannian manifold implies the constancy of the sectional curvature. In [16], Bonome et al. showed that the boundedness of the holomorphic sectional curvature on indefinite almost Hermitian manifolds leads to the space of pointwise constant holomorphic sectional curvature. Therefore in Section 4, we discuss the boundedness of holomorphic sectional curvature of GCR-lightlike submanifolds of an indefinite complex space form. In Section 5, we established a condition for a GCR-lightlike submanifold of an indefinite complex space form to be null holomorphically flat. We also obtain some characterization theorems on holomorphic sectional and holomorphic bisectional curvature.


## 2. Lightlike Submanifolds

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- (see [11]). For a degenerate metric g on M, TM⊥ 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 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
(2.1)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
(2.2)tr⁡(TM)=ltr(TM)⊥S(TM⊥),(2.3)TM-|M=TM⊕tr⁡(TM)=(RadTM⊕ltr(TM))⊥S(TM)⊥S(TM⊥).
For a quasi-orthonormal fields of frames on TM, we have the following.Theorem 2.1 (see [11]).
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
(2.4)g-(Ni,ξj)=δij,g-(Ni,Nj)=0,for  anyi,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 (2.3), the Gauss and Weingarten formulas are given by the following:
(2.5)∇-XY=∇XY+h(X,Y),∇-XU=-AUX+∇X⊥U,
for any X,Y∈Γ(TM) and 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 (2.2) considering the projection morphisms L and S of tr(TM) on ltr(TM) and S(TM⊥), respectively, then Gauss and Weingarten formulas become
(2.6)∇-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), 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,
(2.7)∇-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 (2.6) and (2.7), we obtain
(2.8)g-(hs(X,Y),W)+g-(Y,Dl(X,W))=g(AWX,Y),(2.9)g-(Ds(X,N),W)=g-(AWX,N),
for any X∈Γ(TM), W∈Γ(S(TM⊥)), and N,N′∈Γ(ltr(TM)).LetP be the projection morphism of TM on S(TM), then, using (2.1), we can induce some new geometric objects on the screen distribution S(TM) on M as follows:
(2.10)∇XPY=∇X*PY+h*(X,PY),∇Xξ=-Aξ*X+∇X*tξ,
for any X,Y∈Γ(TM) and ξ∈Γ(RadTM), where {∇X*PY,Aξ*X} and {h*(X,PY), ∇X*tξ} belong to Γ(S(TM)) and Γ(RadTM), respectively. Using (2.6) and (2.10), we obtain
(2.11)g-(hl(X,PY),ξ)=g(Aξ*X,PY),g-(h*(X,PY),N)=g-(ANX,PY),
for any X,Y∈Γ(TM),ξ∈Γ(Rad(TM)) and N∈Γ(ltr(TM)).Denote byR- and R the curvature tensors of ∇- and ∇, respectively, then by straightforward calculations ([11]), we have
(2.12)R-(X,Y)Z=R(X,Y)Z+Ahl(X,Z)Y-Ahl(Y,Z)X+Ahs(X,Z)Y-Ahs(Y,Z)X+(∇Xhl)(Y,Z)-(∇Yhl)(X,Z)+Dl(X,hs(Y,Z))-Dl(Y,hs(X,Z))+(∇Xhs)(Y,Z)-(∇Yhs)(X,Z)+Ds(X,hl(Y,Z))-Ds(Y,hl(X,Z)),
where
(2.13)(∇Xhs)(Y,Z)=∇Xshs(Y,Z)-hs(∇XY,Z)-hs(Y,∇XZ)(2.14)(∇Xhl)(Y,Z)=∇Xlhl(Y,Z)-hl(∇XY,Z)-hl(Y,∇XZ).
Then Codazzi equation is given, respectively, by the following:
(2.15)(R-(X,Y)Z)⊥=(∇Xhl)(Y,Z)-(∇Yhl)(X,Z)+Dl(X,hs(Y,Z))-Dl(Y,hs(X,Z))+(∇Xhs)(Y,Z)-(∇Yhs)(X,Z)+Ds(X,hl(Y,Z))-Ds(Y,hl(X,Z)).
Barros and Romero [17] defined indefinite Kaehler manifolds as follows.Definition 2.2.
Let(M-,J,g-) be an indefinite almost Hermitian manifold and ∇- the Levi-Civita connection on M-, with respect to an indefinite metric g-. Then M- is called an indefinite Kaehler manifold if J is parallel, with respect to ∇-, that is
(2.16)J2=-I,(∇-XJ)Y=0,g-(JX,JY)=g-(X,Y)∀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)={JD2⊕D′}⊥D0,J(D0)=D0,J(D′)=L1⊥L2,where D0 is a nondegenerate distribution on M, L1 and L2 are vector bundle of ltr(TM) and S(TM)⊥, respectively.Then the tangent bundleTM of M is decomposed as TM=D⊥D′, where D=Rad(TM)⊕D0⊕JD2. M is called a proper GCR-lightlike submanifold if D1≠{0},D2≠{0},D0≠{0}, and L2≠{0}. Let Q,P1 and P2 be the projections on D, J(L1)=M1, and J(L2)=M2, respectively. Then, for any X∈Γ(TM), we have
(3.3)X=QX+P1X+P2X,
applying J to (3.3), we obtain
(3.4)JX=TX+wP1X+wP2X,
and we can write (3.4) as follwos:
(3.5)JX=TX+wX,
where TX and wX are the tangential and transversal components of JX, respectively. Similarly,
(3.6)JV=BV+CV,
for any V∈Γ(tr(TM)), where BV and CV are the sections of TM and tr(TM), respectively. Applying J to (3.5) and (3.6), we get
(3.7)T2=-I-Bω,C2=-I-ωB.Differentiating (3.4) and using (2.6), (2.7), and (3.6), we have
(3.8)Ds(X,wP1Y)=-∇XswP2Y+wP2∇XY-hs(X,TY)+Chs(X,Y),(3.9)Dl(X,wP2Y)=-∇XlwP1Y+wP1∇XY-hl(X,TY)+Chl(X,Y).
Using Kaehlerian property of ∇- with (2.7), we have the following lemmas.Lemma 3.2.
LetM be a GCR-lightlike submanifold of an indefinite Kaehlerian manifold M-. Then one has
(3.10)(∇XT)Y=AwYX+Bh(X,Y),(∇Xtw)Y=Ch(X,Y)-h(X,TY),
where X,Y∈Γ(TM) and
(3.11)(∇XT)Y=∇XTY-T∇XY,(∇Xtw)Y=∇XtwY-w∇XY.Lemma 3.3.
LetM be a GCR-lightlike submanifold of an indefinite Kaehlerian manifold M-. Then one has
(3.12)(∇XB)V=ACVX-TAVX,(∇XtC)V=-wAVX-h(X,BV),
where X∈Γ(TM), V∈Γ(tr(TM)), and
(3.13)(∇XB)V=∇XBV-B∇XtV,(∇XtC)V=∇XtCV-C∇XtV.


## 4.  Holomorphic Sectional Curvature of aGCR-Lightlike Submanifold

LetM- be a complex space form of constant holomorphic curvature c. Then the curvature tensor R- of M-(c) is given by the following:
(4.1)R-(X,Y)Z=c4{g-(Y,Z)X-g-(X,Z)Y+g-(JY,Z)JXc4-g-(JX,Z)JY+2g-(X,JY)JZ},
for X,Y,Z vector fields on M-. Using (4.1) and (2.12), we obtain
(4.2)g(R(X,Y)Z,W)=c4{g(Y,Z)g(X,W)-g(X,Z)g(Y,W)+g(JY,Z)g(JX,W)-g(JX,Z)g(JY,W)+2g(X,JY)g(JZ,W)}-g(Ahl(X,Z)Y,W)+g(Ahl(Y,Z)X,W)-g(Ahs(X,Z)Y,W)+g(Ahs(Y,Z)X,W)-g-((∇Xhl)(Y,Z),W)+g-((∇Yhl)(X,Z),W)-g-(Dl(X,hs(Y,Z)),W)+g-(Dl(Y,hs(X,Z)),W).
Using (2.8) in (4.2), we obtain
(4.3)g(R(X,Y)Z,W)=c4{g(Y,Z)g(X,W)-g(X,Z)g(Y,W)+g(JY,Z)g(JX,W)-g(JX,Z)g(JY,W)+2g(X,JY)g(JZ,W)}-g(Ahl(X,Z)Y,W)+g(Ahl(Y,Z)X,W)-g-(hs(Y,W),hs(X,Z))+g-(hs(X,W),hs(Y,Z))-g-((∇Xhl)(Y,Z),W)+g-((∇Yhl)(X,Z),W).
Then the sectional curvature KM(X,Y)=g(R(X,Y)Y,X) of M determined by orthonormal vectors X and Y of Γ(D0⊕M2) and given by the following:
(4.4)KM(X,Y)=c4{1+3g(X,JY)2}-g(Ahl(X,Y)Y,X)+g(Ahl(Y,Y)X,X)-g-(hs(Y,X),hs(X,Y))+g-(hs(X,X),hs(Y,Y)).Corollary 4.1.
LetM be a GCR-lightlike submanifold of an indefinite Complex space form M-(c). Then sectional curvature of M is given by KM(X,Y)=(c/4){1+3g(X,JY)2}, if (i)
M2 defines a totally geodesic foliation in M-,(ii)
D0 defines a totally geodesic foliation in M-,(iii)
M is totally geodesic in M-.Definition 4.2.
The holomorphic sectional curvatureH(X)=g(R(X,JX)JX,X) of M determined by a unit vector X∈Γ(D0) is the sectional curvature of a plane section {X,JX}.Then using (2.11) and (4.4), for a unit vector field X∈Γ(D0), we get
(4.5)H(X)=c-g-(hl(X,JX),h*(JX,X))+g-(hl(JX,JX),h*(X,X))-g-(hs(JX,X),hs(X,JX))+g-(hs(X,X),hs(JX,JX)).
From (3.8), for any X,Y∈Γ(D0), we have
(4.6)hs(X,JY)=wP2∇XY+Chs(X,Y),
and further using (3.7) and (4.6), we have
(4.7)hs(JX,JY)=wP2∇JXY-wBhs(X,Y)-hs(X,Y).
Hence using (4.6) and (4.7) in (4.5), we obtain the expression for holomorphic sectional curvature as follows:
(4.8)H(X)=c-g-(hl(X,JX),h*(JX,X))+g-(hl(JX,JX),h*(X,X))-‖wP2∇XX‖2-‖Chs(X,X)‖2+g-(hs(X,X),wP2∇JXX)+‖Bhs(X,X)‖2-‖hs(X,X)‖2.Theorem 4.3.
LetM be a GCR-lightlike submanifold of an indefinite complex space form M-(c). If M is totally geodesic in M-(c), then H(X)≤c, for any unit vector field X∈Γ(D0).Proof.
Using the hypothesis in (4.8), we get H(X)=c-∥wP2∇XX∥2. Hence the result follows.Theorem 4.44.4 (see [13]).
LetM be a GCR-lightlike submanifold of an indefinite Kaehler manifold M-, then the distribution D is integrable if and only if h(X,JY)=h(Y,JX), for any X,Y∈Γ(D).Theorem 4.5.
LetM be a GCR-lightlike submanifold of an indefinite complex space form M-(c), and D0 is integrable, then H(X)≤c for any unit vector field X∈Γ(D0).Proof.
SinceD0 is integrable therefore using Theorem 4.4, we have h(JX,JY)=-h(X,Y), for any unit vector field X∈Γ(D0). Therefore, from (4.5), we obtain
(4.9)H(X)=c-g-(hl(X,JX),h*(JX,X))-g-(hl(X,X),h*(X,X))-‖hs(X,JX)‖2-‖hs(X,X)‖2≤c.Theorem 4.6.
AGCR-lightlike submanifold of an indefinite complex space form M-(c) is D0-totally geodesic if and only if  (i)
D0 is integrable,(ii)
H(X)=c, for any unit vector field X∈Γ(D0).Proof.
Proof follows from (4.9).Theorem 4.74.7 (see [13]).
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 4.8.
LetM be a GCR-lightlike submanifold of an indefinite complex space form M-(c), and D0 defines a totally geodesic foliation in M, then H(X)≤c, for any unit vector field X∈Γ(D0).Proof.
SinceD0 defines a totally geodesic foliation in M, therefore by definition ∇XX∈Γ(D0), this implies that h*(X,X)=0. Also by using Theorem 4.7, we have Bh(X,X)=0, for any X∈Γ(D0); hence, (4.8) becomes H(X)=c-2∥Chs(X,X)∥2 and the result follows.Definition 4.9.
The horizontal distributionD is called parallel with respect to the induced connection ∇ on M if ∇XY∈D for any X∈Γ(TM) and Y∈Γ(D).Theorem 4.10.
LetM be a GCR-lightlike submanifold of an indefinite complex space form M-(c) and D0 is parallel, with respect to ∇, then H(X)≤c, for any unit vector field X∈Γ(D0).Proof.
SinceD0 is parallel, with respect to the induced connection ∇ on M, therefore ∇X*Y∈Γ(D0) and h*(X,Y)=0, for any X∈Γ(TM) and Y∈Γ(D0). Hence, from (4.8), we obtain H(X)=c+∥Bhs(X,X)∥2-∥hs(X,X)∥2-∥Chs(X,X)∥2, and then by using (3.6), we get H(X)=c-2∥Chs(X,X)∥2. Hence the result is complete.Lemma 4.11.
LetM be a GCR-lightlike submanifold of an indefinite Kaehler manifold M-. If the distribution D0 defines a totally geodesic foliation in M-, then M is D0-geodesic.Proof.
By the definition ofGCR-lightlike submanifold, M is D0-geodesic if g-(hl(X,Y),ξ)=g-(hs(X,Y),W)=0, for any X,Y∈Γ(D0), ξ∈Γ(Rad(TM)), and W∈Γ(S(TM⊥)). Since D0 defines a totally geodesic foliation in M-, therefore g-(hl(X,Y),ξ)=g-(∇-XY,ξ)=0 and g-(hs(X,Y),W)=g-(∇-XY,W)=0. Hence, the assertion follows.Theorem 4.12.
LetM be a GCR-lightlike submanifold of an indefinite complex space form M-(c). If D0 defines a totally geodesic foliation in M-, then H(X)=c, for any unit vector field X∈Γ(D0).Proof.
The result follows directly using Lemma4.11 and (4.8).


## 5. Null Holomorphically FlatGCR-Lightlike Submanifold

Letx∈M- and U be a null vector of TxM-. A plane π of TxM- is called a null plane directed by U if it contains U, g-x(U,V)=0, for any V∈π, and there exists V0∈π such that g-x(V0,V0)≠0. Following Beem-Ehrlich [18], the null sectional curvature of π, with respect to U and ∇-, as a real number, is defined as follows:
(5.1)K-U(π)=g-x(R-(V,U)U,V)g-x(V,V),
where V is an arbitrary non null vector in π. Clearly K-U(π) is independent of V but depends in a quadratic fashion on U.Consideru∈M and a null plane π of TuM directed by ξu∈Rad(TM), then the null sectional curvature of π, with respect to ξu and ∇, as a real number is defined as
(5.2)Kξu(π)=gu(R(Vu,ξu)ξu,Vu)gu(Vu,Vu),
where Vu is an arbitrary non-null vector in π.LetM be a GCR-lightlike submanifold of an indefinite complex space form M-(c) then using (4.3), the null sectional curvature of π, with respect to ξ, is given by the following:
(5.3)Kξ(π)=1g(V,V){g(Ahl(ξ,ξ)V,V)-g(Ahl(V,ξ)ξ,V)+g-(hs(V,V),hs(ξ,ξ))1g(V,V)-g-(hs(ξ,V),hs(V,ξ))-g-((∇Vhl)(ξ,ξ),V)+g-((∇ξhl)(V,ξ),V)}.
Then, using (2.11), we obtain
(5.4)Kξ(π)=1g(V,V){g(h*(V,V),hl(ξ,ξ))-g(h*(ξ,V),hl(V,ξ))+g-(hs(V,V),hs(ξ,ξ))1g(V,V)-g-(hs(ξ,V),hs(V,ξ))-g-((∇Vhl)(ξ,ξ),V)+g-((∇ξhl)(V,ξ),V)}.We know that a planeπ is called holomorphic if it remains invariant under the action of the almost complex structure J-, that is, if π={Z,J-Z}. The sectional curvature associated with the holomorphic plane is called the holomorphic sectional curvature, denoted by H-(π) and given by H-(π)=R-(Z,J-Z,Z,J-Z)/g-(Z,Z)2. The holomorphic plane π={Z,J-Z} is called null or degenerate if and only if Z is a null vector. A manifold (M-,g-,J-) is called null holomorphically flat if the curvature tensor R- satisfies (see [19]). 						(5.5)R-(Z,J-Z,Z,J-Z)=0,
for all null vectors Z. Put g-(R-(X,Y)Z,W)=R-(X,Y,Z,W), then, from (5.4), we obtain
(5.6)R(ξ,J-ξ,ξ,J-ξ)=g(h*(ξ,ξ),hl(J-ξ,J-ξ))-g(h*(J-ξ,ξ),hl(ξ,J-ξ))+g-(hs(ξ,ξ),hs(J-ξ,J-ξ))-g-(hs(J-ξ,ξ),hs(ξ,J-ξ))-g-((∇ξhl)(J-ξ,J-ξ),ξ)+g-((∇J-ξhl)(ξ,J-ξ),ξ).Definition 5.1 (see [20]).
A lightlike submanifold(M,g) of a semi-Riemannian manifold (M-,g-) is said to be a totally umbilical in M- if there is a smooth transversal vector field H∈Γ(tr(TM)) on M, called the transversal curvature vector field of M, such that, for X,Y∈Γ(TM),
(5.7)h(X,Y)=Hg-(X,Y).
Using (2.6), it is clear that M is a totally umbilical if and only if on each coordinate neighborhood u there exist smooth vector fields Hl∈Γ(ltr(TM)) and Hs∈Γ(S(TM⊥)) such that
(5.8)hl(X,Y)=Hlg(X,Y),hs(X,Y)=Hsg(X,Y),Dl(X,W)=0,
for X,Y∈Γ(TM) and W∈Γ(S(TM⊥)). A lightlike submanifold is said to be totally geodesic if h(X,Y)=0, for any X,Y∈Γ(TM). Therefore, in other words, a lightlike submanifold is totally geodesic if Hl=0 and Hs=0.
LetM be a totally umbilical lightlike submanifold, then, using above definition, we have h(J-ξ,J-ξ)=Hg(J-ξ,J-ξ)=Hg(ξ,ξ)=0 and h(ξ,J-ξ)=Hg(ξ,J-ξ)=0, for any ξ∈Γ(Rad(TM)). Thus, from (5.6), we have the following theorem.Theorem 5.2.
LetM be a GCR-lightlike submanifold of an indefinite complex space form M-(c). If M is totally umbilical lightlike submanifold in M-(c), then M is null holomorphically flat.Moreover, from (5.6), it is clear that the expression of R(ξ,J-ξ,ξ,J-ξ) is expressed in terms of screen second fundamental forms of M, thus GCR-lightlike submanifold M of an indefinite complex space form M-(c) is null holomorphically flat if M is totally geodesic.


## 6. Holomorphic Bisectional Curvature of aGCR-Lightlike Submanifold

Definition 6.1.
The holomorphic bisectional for the pair of unit vector fields{X,Y} on M- is given by H-(X,Y)=g-(R-(X,JX)JY,Y).Theorem 6.2.
LetM be a mixed totally geodesic GCR-lightlike submanifold of an indefinite Kaehler manifold M- with D0 parallel distribution. Then H-(X,Z)=0, for any unit vector fields X∈Γ(D0) and Z∈Γ(M2).Proof.
LetX,Y∈Γ(D0) and Z∈Γ(M2) then, by using that hypothesis that the distribution D0 is a parallel, with respect to ∇ on M, we have g(T∇XZ,Y)=-g-(∇-XZ,TY)=g(Z,∇XTY)=0. Hence, the non degeneracy of the distribution D0 implies that
(6.1)∇XZ∈Γ(D′),
for anyZ∈Γ(M2). Now replacing Y by JX, respectively, in (2.15) and then taking inner product with JZ, for any X∈Γ(D0) and Z∈Γ(M2), we get
(6.2)H-(X,Z)=g-(hs(∇XJX,Z),JZ)-g-(∇Xhs(JX,Z),JZ)+g-(hs(JX,∇XZ),JZ)+g-(∇JXhs(X,Z),JZ)-g-(hs(∇JXX,Z),JZ)-g-(hs(X,∇JXZ),JZ)-g-(Ds(X,hl(JX,Z)),JZ)+g-(Ds(JX,hl(X,Z)),JZ).
Hence by using that M is mixed totally geodesic with (6.1), the assertion follows.Theorem 6.3.
In order that an indefinite complex space formM-(c) may admit a mixed totally geodesic GCR-lightlike submanifold M with parallel horizontal distribution D0, it is necessary that c=0.Proof.
LetX∈Γ(D0) and Z∈Γ(M2) be unit vector fields, then (4.1) implies that H-(X,Z)=-(c/2)g(X,X)g(Z,Z), then the non degeneracy of the distributions D0 and M2 with the Theorem 6.2, we obtain c=0. Hence, the result follows.Lemma 6.4.
LetM be a GCR-lightlike submanifold of an indefinite Kaehler manifold M-. Then we have the following:(i)
ifD′ defines a totally geodesic foliation in M- then g-(hs(D′,D0),JD′)=0,(ii)
ifD0 is a parallel distribution, with respect to ∇, then h(Z,JX)=Ch(Z,X) for any X∈Γ(D0),Z∈Γ(M2).Proof.
(i) LetD′ define a totally geodesic foliation in M- this implies that ∇-XY=∇XY∈Γ(D′) and h(X,Y)=0, for any X,Y∈Γ(D′). Therefore by using (3.10), we obtain AwYX=-Bh(X,Y)=0. Let Z∈Γ(D0), then by using (2.8), we get 0=g(AwYX,Z)=g-(hs(X,Z),wY). Thus we have g-(hs(D′,D0),JD′)=0.
(ii) LetD0 is a parallel distribution with respect to the induced connection ∇, therefore ∇XY∈Γ(D0), for any Y∈Γ(D0),X∈Γ(TM). Since M- is Kaehler manifold, therefore for Z∈Γ(M2) and X∈Γ(D0), we have ∇-ZJX=J∇-ZX. This implies that ∇ZJX+h(Z,JX)=J∇ZX+Bh(Z,X)+Ch(Z,X), then by equating transversal components on both sides, we get the result.Theorem 6.5.
LetM be a GCR-lightlike submanifold of an indefinite Kaehler manifold M-. If D0 is parallel with respect to the induced connection ∇, and M2 defines a totally geodesic foliation in M-, then
(6.3)H-(X,Z)=g-(h(JX,Z),J∇XZ)-g-(h(X,Z),J∇JXZ)+g(AJZJX,∇XZ)-g(AJZX,∇JXZ)+2‖Chs(X,Z)‖2,
for any unit vector fields X∈Γ(D0) and Z∈Γ(M2).Proof.
LetX∈Γ(D0) and Z∈Γ(M2), then the equation of Codazzi (2.15) becomes
(6.4)g-(R-(X,JX)Z,JZ)=g-((∇Xhs)(JX,Z),JZ)-g-((∇JXhs)(X,Z),JZ)+g-(Ds(X,hl(JX,Z)),JZ)-g-(Ds(JX,hl(X,Z)),JZ).
By using (2.13) and (6.1) with the Lemma 6.4 (i), we obtain
(6.5)H-(X,Z)=g-(∇JXshs(X,Z),JZ)-g-(∇Xshs(JX,Z),JZ)+g(AJZJX,∇XZ)-g(AJZX,∇JXZ)-g-(Ds(X,hl(JX,Z)),JZ)+g-(Ds(JX,hl(X,Z)),JZ).
Now by using (2.7) with the Lemma 6.4 (ii), we have
(6.6)g-(∇Xshs(JX,Z),JZ)=g-(∇-Xhs(JX,Z),JZ)=-g-(hs(JX,Z),J∇XZ)-∥Chs(X,Z)∥2,
and similarly
(6.7)g-(∇JXshs(X,Z),JZ)=-g-(hs(X,Z),J∇JXZ)+‖Chs(X,Z)‖2.
By using (2.7), we have
(6.8)g-(Ds(X,hl(JX,Z)),JZ)=-g-(hl(JX,Z),J∇XZ),(6.9)g-(Ds(JX,hl(X,Z)),JZ)=-g-(hl(X,Z),J∇JXZ).
Hence by using (6.6)–(6.9) in (6.5), the result follows.Definition 6.66.6 (see [13]).
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.Theorem 6.7 (see [14]).
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′).Theorem 6.8.
LetM be a GCR-lightlike submanifold of an indefinite Kaehler manifold M-. If (∇XT)Y=0, for any X,Y∈Γ(TM), then
(6.10)H-(X,Z)=2g-(hs(JX,Z),Jhs(X,Z)),
for any unit vector fields X∈Γ(D0) and Z∈Γ(M2).Proof.
Let(∇XT)Y=0, then (3.10) implies that
(6.11)AwYX+Bh(X,Y)=0,
for any X,Y∈Γ(TM). Let Y∈Γ(D),X∈Γ(TM), then (6.11) gives
(6.12)Bh(X,Y)=0.
Let Z∈Γ(D′) and X∈Γ(D), then using (6.11) and (6.12), we obtain
(6.13)AwZX=0.
Particularly choosing X∈Γ(D0) and Z∈Γ(M2) in (2.15), we get
(6.14)g-(R-(X,JX)Z,JZ)=g-((∇Xhs)(JX,Z),JZ)-g-((∇JXhs)(X,Z),JZ)+g-(Ds(X,hl(JX,Z)),JZ)-g-(Ds(JX,hl(X,Z)),JZ).
Since (∇XT)Y=0 therefore, by using Theorem 6.7, the distributions D and D′ define totally geodesic foliations in M. Then D′ defines totally geodesic foliation in M implies that for any Z1,Z2∈Γ(D′), we have ∇Z1Z2∈Γ(D′). Therefore, using (3.10) and (3.11), we get AwZ2Z1+Bh(Z1,Z2)=0. By taking inner product with X∈Γ(D0) and using (2.8), we get
(6.15)g-(hs(Z1,X),wZ2)=0.
Also, (3.11) implies that T∇XZ=0, that is, ∇XZ∈Γ(D′). Therefore using (2.8), (2.13), (6.13), and (6.15) in (6.14), we obtain
(6.16)g-(R-(X,JX)Z,JZ)=g-(∇Xshs(JX,Z),JZ)-g-(∇JXshs(X,Z),JZ)+g-(Ds(X,hl(JX,Z)),JZ)-g-(Ds(JX,hl(X,Z)),JZ).
Now using (2.6), (2.7), (2.8), and (6.15), we have
(6.17)g-(∇Xshs(JX,Z),JZ)=g-(Jhs(JX,Z),hs(X,Z)).
Similarly,
(6.18)g-(∇JXshs(X,Z),JZ)=-g-(Jhs(JX,Z),hs(X,Z)).
Also using (2.7), we have
(6.19)g-(Ds(X,hl(JX,Z)),JZ)=0,g-(Ds(JX,hl(X,Z)),JZ)=0.
Hence, using (6.17)–(6.19) in (6.16), the result follows.Lemma 6.9.
LetM be a GCR-lightlike submanifold of an indefinite Kaehler manifold such that (∇XB)Y=0. Then hs(JX,Z)∈Γ(L2⊥) for any X∈Γ(D0), Z∈Γ(M2).Proof.
Since(∇XB)Y=0, therefore, from (3.12) we have TAWX=0, for any X∈Γ(TM) and W∈Γ(L2), this implies that
(6.20)AWX∈Γ(D′),
for any W∈Γ(L2) and X∈Γ(TM). Since hs(JX,Z)∈Γ(S(TM⊥)), therefore, to prove that hs(JX,Z)∈Γ(L2⊥), it is sufficient to prove that g-(hs(JX,Z),W)=0, for any W∈Γ(L2). Let X∈Γ(D0) and Z∈Γ(M2) such that W=wZ, we have g(∇UZ,X)=g-(∇-UZ,X)=g-(∇-UJZ,JX)=-g(AJZU,JX)=g(JAJZU,X), then using (2.8) we obtain
(6.21)g-(hs(JX,Z),W)=-g(JAWZ,X)=-g(∇ZZ,X)=-g(T∇ZZ,TX).
Since, from (3.10), we have T∇ZZ=-AwZZ-Bh(Z,Z), then using (6.20) in (6.21), the result follows.Theorem 6.10.
LetM be a GCR-lightlike submanifold of an indefinite Kaehler manifold such that (∇XB)Y=0, then
(6.22)H-(X,Z)=2g-(Jhs(X,Z),hs(JX,Z))-g(AJZ∇JXX,Z)+g(AJZ∇XJX,Z),
for any unit vector fields X∈Γ(D0) and Z∈Γ(M2).Proof.
LetX∈Γ(D0) and Z∈Γ(M2), then, from (2.15) and (2.13), we obtain
(6.23)g-(R-(X,JX)Z,JZ)=g-(∇Xshs(JX,Z),JZ)-g-(hs(∇XJX,Z),JZ)-g-(hs(JX,∇XZ),JZ)-g-(∇JXshs(X,Z),JZ)+g-(hs(∇JXX,Z),JZ)+g-(hs(X,∇JXZ),JZ)+g-(Ds(X,hl(JX,Z)),JZ)-g-(Ds(JX,hl(X,Z)),JZ),
using (6.20) in (2.9), we obtain
(6.24)g-(Ds(X,hl(JX,Z)),JZ)=0,g-(Ds(JX,hl(X,Z)),JZ)=0.
Now consider
(6.25)g-(∇Xshs(JX,Z),JZ)=-g-(hs(JX,Z),J∇-XZ)=g-(hs(X,Z),Jhs(JX,Z)),
and similarly
(6.26)g-(∇JXshs(X,Z),JZ)=-g-(hs(X,Z),Jhs(JX,Z)).
Also using (2.8), we have
(6.27)g-(hs(∇XJX,Z),JZ)=g(AJZ∇XJX,Z),g-(hs(∇JXX,Z),JZ)=g(AJZ∇JXX,Z).
Using (6.20) in (2.8), we have
(6.28)g-(hs(JX,∇XZ),JZ)=0,g-(hs(X,∇JXZ),JZ)=0.
Thus using (6.24)–(6.28) in (6.23), the result follows.Theorem 6.11.
LetM be a mixed foliate GCR-lightlike submanifold of an indefinite Kaehler manifold M-, and S(TM) is parallel distribution, with respect to the induced connection ∇, then
(6.29)H-(X,Z)=2g-(AJZJX,JAJZX),
for any unit vector fields X∈Γ(D0) and Z∈Γ(M2).Proof.
SinceM is mixed foliate therefore for any X∈Γ(D0) and Z∈Γ(M2), Codazzi equation (2.15) and (2.13), imply that
(6.30)H-(X,Z)=g-(hs(∇XJX,Z),JZ)+g-(hs(JX,∇XZ),JZ)-g-(hs(∇JXX,Z),JZ)-g-(hs(X,∇JXZ),JZ).
Since using (2.8) and the hypothesis, we have g(AJZX,JW)=0 and g(AJZX,Jξ)=g-(hs(X,Jξ),JZ)=g-(∇-XJξ,JZ)=-g(ξ,∇XZ)=0. Therefore, by the definition of a GCR-lightlike submanifold, we have
(6.31)AJZX∈Γ(D).
Thus, using (2.7), (2.8), and (6.31) with the hypothesis, we obtain
(6.32)g-(hs(JX,∇XZ),JZ)=g-(AJZJX,∇-XZ)=g-(JAJZJX,J∇-XZ)=g-(AJZJX,JAJZX)-g-(AJZJX,JDl(X,JZ))=g-(AJZJX,JAJZX).
Similarly, we obtain
(6.33)g-(hs(X,∇JXZ),JZ)=g-(AJZX,JAJZJX),g-(hs(∇JXX,Z),JZ)=g(AJZZ,∇JXX)-g-(∇JXX,Dl(JZ,Z)),g-(hs(∇XJX,Z),JZ)=g(AJZZ,∇XJX)-g-(∇XJX,Dl(JZ,Z)).
Thus, (6.30) becomes
(6.34)H-(X,Z)=2g-(AJZJX,JAJZX)-g(AJZZ,∇JXX)+g-(∇JXX,Dl(JZ,Z))+g(AJZZ,∇XJX)-g-(∇XJX,Dl(JZ,Z)).
Since the distribution D is integrable, therefore ∇JXX-∇XJX=[X,JX]=X′∈Γ(D), then (6.34) becomes H-(X,Z)=2g-(AJZJX,JAJZX)-g(AJZZ,X′)+g-(X′,Dl(JZ,Z)). Hence, using the hypothesis and (2.8), the result follows.



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*Source: 356263-2012-07-03.xml*