Characterization of Holomorphic Bisectional Curvature of GCR-Lightlike Submanifolds

(1)

Volume 2012, Article ID 356263,18pages doi:10.1155/2012/356263

Research Article

Characterization of Holomorphic Bisectional Curvature of GCR-Lightlike Submanifolds

Sangeet Kumar,

¹

Rakesh Kumar,

²

and R. K. Nagaich

³

1Department of Applied Sciences, Rayat Institute of Engineering & Information Technology, Railmajra, SBS Nagar, Punjab 144533, India

2Department of Basic and Applied Sciences, University College of Engineering, Punjabi University, Patiala 147002, India

3Department of Mathematics, Punjabi University, Patiala 147002, India

Correspondence should be addressed to Rakesh Kumar,dr [email protected] Received 27 March 2012; Accepted 19 May 2012

Academic Editor: M. Lakshmanan

Copyrightq2012 Sangeet Kumar et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We obtain the expressions for sectional curvature, holomorphic sectional curvature, and holomorphic bisectional curvature of a GCR-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.

1. Introduction

The study of CR submanifolds of Kaehler manifolds was initiated by Bejancu 1, as a generalization of totally real and complex submanifolds and further developed by2–7. The CR structures on real hypresurfaces of complex manifolds have interesting applications to relativity. Penrose8discovered 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 aCRmanifold. Duggal9,10studied 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

(2)

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 betweenCRandSCRcases. Later on, Duggal and Sahin 13introducedGCR-lightlike submanifolds of indefinite Kaehler manifolds, which behaves as an umbrella of invariantcomplex, screen real and CR-lightlike submanifolds and also studied the existenceor nonexistenceof this new class in an indefinite space form. R. Kumar et al.14studied geodesicGCR-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 a GCR-lightlike submanifold of an indefinite Kaehler manifold. In15, Kulkarni showed that the boundedness of the sectional curvature on a semi-Riemannian manifold implies the constancy of the sectional curvature. In16, 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 inSection 4, we discuss the boundedness of holomorphic sectional curvature ofGCR-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 realmn-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 ofg onM. Ifg is degenerate on the tangent bundleTM ofM, 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 T_xM. Thus both T_xM and T_xM^⊥ are degenerate orthogonal subspaces but no longer complementary. In this case, there exists a subspace RadTxMTxM∩TxM^⊥which is known as radicalnullsubspace. If the mapping RadTM :x ∈ M → RadT_xMdefines a smooth distribution onMof rankr > 0, then the submanifoldMofMis called anr-lightlike submanifold and RadTMis called the radical distribution onM.

Screen distribution STM is a semi-Riemannian complementary distribution of RadTMinTM, that is

TMRadTM⊥STM, 2.1

(3)

and STM^⊥ is a complementary vector subbundle to RadTM in TM^⊥. Let trTM and ltrTM be complementary but not orthogonal vector bundles to TM in TM|M and to RadTMinSTM^⊥^⊥, respectively. Then we have

trTM ltrTM⊥S TM^⊥

, 2.2

TM|MTM⊕trTM RadTM⊕ltrTM⊥STM⊥S TM^⊥

. 2.3

For a quasi-orthonormal fields of frames onTM, we have the following.

Theorem 2.1 see11. Let M, g, STM, STM^⊥ be an r-lightlike submanifold of a semi- Riemannian manifoldM, g. Then there exists a complementary vector bundle ltrTMof RadTM inSTM^⊥^⊥and a basis ofΓltrTM|uconsisting of smooth section{Ni}ofSTM^⊥^⊥|u, whereu is a coordinate neighborhood ofMsuch that

g Ni, ξj

δij, g

Ni, Nj

0, for anyi, j ∈ {1,2, . . . , r}, 2.4 where{ξ1, . . . , ξr}is a lightlike basis ofΓRadTM.

Let∇be the Levi-Civita connection onM, then, according to the decomposition2.3, the Gauss and Weingarten formulas are given by the following:

∇XY ∇XY hX, Y, ∇XU−AUX∇^⊥_XU, 2.5

for anyX, Y ∈ΓTMandU ∈ΓtrTM, where{∇XY, A_UX}and{hX, Y,∇^⊥_XU}belong toΓTMandΓtrTM, respectively. Here∇is a torsion-free linear connection onM,his a symmetric bilinear form onΓTMwhich is called second fundamental form, andAUis a linear a operator onMand known as shape operator.

According to 2.2 considering the projection morphisms L and S of trTM on ltrTMandSTM^⊥, respectively, then Gauss and Weingarten formulas become

∇XY ∇XYh^lX, Y h^sX, Y, ∇XU−AUXD^l_XUD^s_XU, 2.6

where we put h^lX, Y LhX, Y, h^sX, Y ShX, Y, D_X^lU L∇^⊥_XU, D_X^sU S∇^⊥_XU. Ash^landh^sareΓltrTMvalued andΓSTM^⊥valued, respectively, therefore they are called the lightlike second fundamental form and the screen second fundamental form onM. In particular,

∇XN−ANX∇^l_XND^sX, N, ∇XW−AWX∇^s_XWD^lX, W, 2.7

whereX∈ΓTM, N∈ΓltrTM, andW ∈ΓSTM^⊥. Using2.6and2.7, we obtain gh^sX, Y, W g

Y, D^lX, W

gAWX, Y, 2.8 gD^sX, N, W gAWX, N, 2.9 for anyX∈ΓTM,W∈ΓSTM^⊥, andN, N∈ΓltrTM.

(4)

LetP be the projection morphism ofTMonSTM, then, using2.1, we can induce some new geometric objects on the screen distributionSTMonMas follows:

∇XPY ∇^∗_XPYh^∗X, PY, ∇Xξ−A^∗_ξX∇^∗t_Xξ, 2.10

for anyX, Y ∈ΓTMandξ∈ΓRadTM, where{∇^∗_XPY, A^∗_ξX}and{h^∗X, PY,∇^∗t_Xξ}belong toΓSTMandΓRadTM, respectively. Using2.6and2.10, we obtain

g

h^lX, PY, ξ g

A^∗_ξX, PY

, gh^∗X, PY, N gANX, PY, 2.11

for anyX, Y ∈ΓTM, ξ∈ΓRadTMandN∈ΓltrTM.

Denote byRandRthe curvature tensors of∇and∇, respectively, then by straightfor- ward calculations11, we have

RX, YZRX, YZA_h^l_X,ZY −A_h^l_Y,ZXA_h^s_X,ZY

−Ah^sY,ZX

∇Xh^l

Y, Z−

∇Yh^l X, Z D^lX, h^sY, Z−D^lY, h^sX, Z ∇Xh^sY, Z

−∇Yh^sX, Z D^s

X, h^lY, Z

−D^s

Y, h^lX, Z ,

2.12

where

∇Xh^sY, Z ∇^s_Xh^sY, Z−h^s∇XY, Z−h^sY,∇XZ 2.13 ∇Xh^l

Y, Z ∇^l_Xh^lY, Z−h^l∇XY, Z−h^lY,∇XZ. 2.14

Then Codazzi equation is given, respectively, by the following:

RX, YZ_⊥

∇Xh^l

Y, Z−

∇Yh^l X, Z

D^lX, h^sY, Z−D^lY, h^sX, Z ∇Xh^sY, Z

−∇Yh^sX, Z D^s

X, h^lY, Z

−D^s

Y, h^lX, Z .

2.15

Barros and Romero17defined indefinite Kaehler manifolds as follows.

Definition 2.2. LetM, J, gbe 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 ifJis parallel, with respect to∇, that is

J² −I,

∇XJ

Y 0, gJX, JY gX, Y ∀X, Y ∈Γ TM

. 2.16

(5)

3. Generalized Cauchy-Riemann Lightlike Submanifolds

Definition 3.1. LetM, g, STMbe a real lightlike submanifold of an indefinite Kaehler man- ifoldM, g, J, thenMis called a generalized Cauchy-RiemannGCR-lightlike submanifold if the following conditions are satisfied.

AThere exist two subbundlesD₁andD₂of RadTMsuch that

RadTM D1⊕D2, JD1 D1, JD2⊂STM. 3.1

BThere exist two subbundlesD₀andDofSTMsuch that

STM

JD2⊕D

⊥D0, JD0 D0, J D

L1⊥L2, 3.2

whereD₀is a nondegenerate distribution onM,L₁andL₂are vector bundle of ltrTMand STM^⊥, respectively.

Then the tangent bundle TM of M is decomposed as TM D ⊥ D, where D RadTM ⊕ D0 ⊕ JD2. M is called a proper GCR-lightlike submanifold if D₁/{0}, D2/{0}, D0/{0}, andL₂/{0}. LetQ, P₁andP₂be the projections onD,JL1 M₁, andJL2 M₂, respectively. Then, for anyX ∈ΓTM, we have

XQXP₁XP₂X, 3.3

applyingJto3.3, we obtain

JXTXwP₁XwP₂X, 3.4

and we can write3.4as follwos:

JXTXwX, 3.5

whereTX andwX are the tangential and transversal components ofJX, respectively. Simi- larly,

JV BVCV, 3.6

for anyV ∈ΓtrTM, whereBV andCV are the sections ofTMand trTM, respectively.

ApplyingJto3.5and3.6, we get

T²−I−Bω, C²−I−ωB. 3.7

(6)

Diﬀerentiating3.4and using2.6,2.7, and3.6, we have

D^sX, wP₁Y −∇^s_XwP₂YwP₂∇XY−h^sX, TY Ch^sX, Y, 3.8 D^lX, wP2Y −∇^l_XwP₁YwP₁∇XY−h^lX, TY Ch^lX, Y. 3.9

Using Kaehlerian property of∇with2.7, we have the following lemmas.

Lemma 3.2. LetMbe aGCR-lightlike submanifold of an indefinite Kaehlerian manifoldM. Then one has

∇XTY AwYXBhX, Y,

∇^t_Xw

Y ChX, Y−hX, TY, 3.10

whereX, Y ∈ΓTMand

∇XTY ∇XTY−T∇XY,

∇^t_Xw

Y ∇^t_XwY−w∇XY. 3.11

Lemma 3.3. LetMbe aGCR-lightlike submanifold of an indefinite Kaehlerian manifoldM. Then one has

∇XBV ACVX−TAVX,

∇^t_XC

V −wAVX−hX, BV, 3.12

whereX∈ΓTM,V ∈ΓtrTM, and

∇XBV ∇XBV−B∇^t_XV,

∇^t_XC

V ∇^t_XCV −C∇^t_XV. 3.13

4. Holomorphic Sectional Curvature of a GCR-Lightlike Submanifold

Let Mbe a complex space form of constant holomorphic curvaturec. Then the curvature tensorRofMcis given by the following:

RX, YZ c 4

gY, ZX−gX, ZYgJY, ZJX

−gJX, ZJY2gX, JYJZ ,

4.1

(7)

forX, Y, Zvector fields onM. Using4.1and2.12, we obtain gRX, YZ, W c

4

gY, ZgX, W−gX, ZgY, W gJY, ZgJX, W

−gJX, ZgJY, W 2gX, JYgJZ, W

−g

A_h^l_X,ZY, W g

A_h^l_Y,ZX, W

−g

A_h^s_X,ZY, W g

A_h^s_Y,ZX, W

−g

∇Xh^l

Y, Z, W g

∇Yh^l

X, Z, W

−g

D^lX, h^sY, Z, W g

D^lY, h^sX, Z, W .

4.2

Using2.8in4.2, we obtain gRX, YZ, W c

4

gY, ZgX, W−gX, ZgY, W gJY, ZgJX, W

−gJX, ZgJY, W 2gX, JYgJZ, W

−g

A_h^l_X,ZY, W g

A_h^l_Y,ZX, W

−gh^sY, W, h^sX, Z gh^sX, W, h^sY, Z

−g

∇Xh^l

Y, Z, W g

∇Yh^l

X, Z, W .

4.3

Then the sectional curvatureKMX, Y gRX, YY, XofMdetermined by orthonormal vectorsXandY ofΓD0⊕M₂and given by the following:

KMX, Y c 4

13gX, JY² −g

A_h^l_X,YY, X g

A_h^l_Y,YX, X

−gh^sY, X, h^sX, Y gh^sX, X, h^sY, Y.

4.4

Corollary 4.1. LetMbe aGCR-lightlike submanifold of an indefinite Complex space formMc.

Then sectional curvature ofMis given byKMX, Y c/4{13gX, JY²}, if iM₂defines a totally geodesic foliation inM,

iiD0defines a totally geodesic foliation inM, iiiMis totally geodesic inM.

Definition 4.2. The holomorphic sectional curvatureHX gRX, JXJX, XofMdeter- mined by a unit vectorX ∈ΓD0is the sectional curvature of a plane section{X, JX}.

Then using2.11and4.4, for a unit vector fieldX∈ΓD0, we get HX c−g

h^lX, JX, h^∗JX, X g

h^lJX, JX, h^∗X, X

−gh^sJX, X, h^sX, JX gh^sX, X, h^sJX, JX.

4.5

(8)

From3.8, for anyX, Y ∈ΓD0, we have

h^sX, JY wP₂∇XY Ch^sX, Y, 4.6

and further using3.7and4.6, we have

h^sJX, JY wP₂∇JXY−wBh^sX, Y−h^sX, Y. 4.7

Hence using 4.6 and 4.7 in 4.5, we obtain the expression for holomorphic sectional curvature as follows:

HX c−g

h^lX, JX, h^∗JX, X g

h^lJX, JX, h^∗X, X

− wP₂∇XX²

− Ch^sX, X²g

h^sX, X, wP2∇JXX

Bh^sX, X²− h^sX, X².

4.8

Theorem 4.3. LetMbe aGCR-lightlike submanifold of an indefinite complex space formMc. If Mis totally geodesic inMc, thenHX≤c, for any unit vector fieldX ∈ΓD0.

Proof. Using the hypothesis in4.8, we getHX c−wP2∇XX². Hence the result follows.

Theorem 4.4see13. LetMbe aGCR-lightlike submanifold of an indefinite Kaehler manifold M, then the distributionDis integrable if and only ifhX, JY hY, JX, for anyX, Y ∈ΓD.

Theorem 4.5. LetMbe aGCR-lightlike submanifold of an indefinite complex space formMc, and D0is integrable, thenHX≤cfor any unit vector fieldX∈ΓD0.

Proof. SinceD₀is integrable therefore usingTheorem 4.4, we havehJX, JY −hX, Y, for any unit vector fieldX∈ΓD0. Therefore, from4.5, we obtain

HX c−g

h^lX, JX, h^∗JX, X

−g

h^lX, X, h^∗X, X

− h^sX, JX²− h^sX, X²≤c.

4.9

Theorem 4.6. AGCR-lightlike submanifold of an indefinite complex space formMcisD0-totally geodesic if and only if

iD₀is integrable,

iiHX c, for any unit vector fieldX ∈ΓD0. Proof. Proof follows from4.9.

(9)

Theorem 4.7see13. LetMbe aGCR-lightlike submanifold of an indefinite Kaehler manifold M. Then the distributionDdefines a totally geodesic foliation inMif and only ifBhX, Y 0, for anyX, Y ∈ΓD.

Theorem 4.8. LetMbe aGCR-lightlike submanifold of an indefinite complex space formMc, and D₀defines a totally geodesic foliation inM, thenHX≤c, for any unit vector fieldX∈ΓD0. Proof. SinceD₀defines a totally geodesic foliation inM, therefore by definition∇XX ∈ΓD0, this implies that h^∗X, X 0. Also by usingTheorem 4.7, we haveBhX, X 0, for any X∈ΓD0; hence,4.8becomesHX c−2Ch^sX, X²and the result follows.

Definition 4.9. The horizontal distribution D is called parallel with respect to the induced connection∇onMif∇XY ∈Dfor anyX∈ΓTMandY ∈ΓD.

Theorem 4.10. LetMbe aGCR-lightlike submanifold of an indefinite complex space formMcand D0is parallel, with respect to∇, thenHX≤c, for any unit vector fieldX ∈ΓD0.

Proof. SinceD₀is parallel, with respect to the induced connection∇onM, therefore∇^∗_XY ∈ ΓD0andh^∗X, Y 0, for anyX ∈ ΓTMandY ∈ ΓD0. Hence, from 4.8, we obtain HX cBh^sX, X²−h^sX, X²−Ch^sX, X², and then by using3.6, we getHX c−2Ch^sX, X². Hence the result is complete.

Lemma 4.11. LetM be aGCR-lightlike submanifold of an indefinite Kaehler manifold M. If the distributionD₀defines a totally geodesic foliation inM, thenMisD₀-geodesic.

Proof. By the definition ofGCR-lightlike submanifold, MisD0-geodesic ifgh^lX, Y, ξ gh^sX, Y, W 0, for anyX, Y ∈ ΓD0,ξ ∈ ΓRadTM, andW ∈ ΓSTM^⊥. Since D₀ defines a totally geodesic foliation in M, thereforegh^lX, Y, ξ g∇XY, ξ 0 and gh^sX, Y, W g∇XY, W 0. Hence, the assertion follows.

Theorem 4.12. LetMbe aGCR-lightlike submanifold of an indefinite complex space formMc. If D₀defines a totally geodesic foliation inM, thenHX c, for any unit vector fieldX∈ΓD0.

Proof. The result follows directly usingLemma 4.11and4.8.

5. Null Holomorphically Flat GCR-Lightlike Submanifold

Letx∈MandUbe a null vector ofTxM. A planeπofTxMis called a null plane directed by Uif it containsU,g_xU, V 0, for anyV ∈π, and there existsV0∈πsuch thatg_xV0, V0/0.

Following Beem-Ehrlich18, the null sectional curvature ofπ, with respect toUand∇, as a real number, is defined as follows:

KUπ g_x

RV, UU, V

g_xV, V , 5.1

whereVis an arbitrary non null vector inπ. ClearlyKUπis independent ofV but depends in a quadratic fashion onU.

(10)

Consideru∈Mand a null planeπ ofTuMdirected byξu ∈RadTM, then the null sectional curvature ofπ, with respect toξ_uand∇, as a real number is defined as

Kξuπ guRVu, ξuξu, Vu

g_uVu, V_u , 5.2

whereV_uis an arbitrary non-null vector inπ.

LetMbe aGCR-lightlike submanifold of an indefinite complex space formMcthen using4.3, the null sectional curvature ofπ, with respect toξ, is given by the following:

Kξπ 1 gV, V

g

A_h^l_ξ,ξV, V

−g

A_h^l_V,ξξ, V

gh^sV, V, h^sξ, ξ

−gh^sξ, V, h^sV, ξ−g

∇Vh^l

ξ, ξ, V g

∇ξh^l

V, ξ, V . 5.3

Then, using2.11, we obtain

Kξπ 1 gV, V

g

h^∗V, V, h^lξ, ξ

−g

h^∗ξ, V, h^lV, ξ

gh^sV, V, h^sξ, ξ

−gh^sξ, V, h^sV, ξ−g

∇Vh^l

ξ, ξ, V g

∇ξh^l

V, ξ, V . 5.4

We know that a planeπ is called holomorphic if it remains invariant under the action of the almost complex structureJ, that is, ifπ {Z, JZ}. The sectional curvature associated with the holomorphic plane is called the holomorphic sectional curvature, denoted byHπ and given by Hπ RZ, JZ, Z, JZ/gZ, Z². The holomorphic plane π {Z, JZ} is called null or degenerate if and only ifZis a null vector. A manifoldM, g, Jis called null holomorphically flat if the curvature tensorRsatisfiessee19.

R

Z, JZ, Z, JZ

0, 5.5

for all null vectorsZ. PutgRX, YZ, W RX, Y, Z, W, then, from5.4, we obtain R

ξ, Jξ, ξ, Jξ g

h^∗ξ, ξ, h^l

Jξ, Jξ

−g h^∗

Jξ, ξ , h^l

ξ, Jξ g

h^sξ, ξ, h^s

Jξ, Jξ

−g h^s

Jξ, ξ , h^s

ξ, Jξ

−g

∇ξh^l Jξ, Jξ

, ξ g

∇_Jξh^l ξ, Jξ

, ξ .

5.6

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

(11)

H ∈ ΓtrTM on M, called the transversal curvature vector field of M, such that, for X, Y ∈ΓTM,

hX, Y HgX, Y. 5.7

Using 2.6, it is clear that M is a totally umbilical if and only if on each coordinate neighborhooduthere exist smooth vector fieldsH^l∈ΓltrTMandH^s∈ΓSTM^⊥such that

h^lX, Y H^lgX, Y, h^sX, Y H^sgX, Y, D^lX, W 0, 5.8

forX, Y∈ΓTMandW∈ΓSTM^⊥. A lightlike submanifold is said to be totally geodesic ifhX, Y 0, for anyX, Y ∈ ΓTM. Therefore, in other words, a lightlike submanifold is totally geodesic ifH^l0 andH^s0.

LetMbe a totally umbilical lightlike submanifold, then, using above definition, we have hJξ, Jξ HgJξ, Jξ Hgξ, ξ 0 and hξ, Jξ Hgξ, Jξ 0, for any ξ∈ΓRadTM. Thus, from5.6, we have the following theorem.

Theorem 5.2. LetMbe aGCR-lightlike submanifold of an indefinite complex space formMc. If Mis totally umbilical lightlike submanifold inMc, thenMis null holomorphically flat.

Moreover, from 5.6, it is clear that the expression of Rξ, Jξ, ξ, Jξ is expressed in terms of screen second fundamental forms ofM, thusGCR-lightlike submanifoldMof an indefinite complex space formMcis null holomorphically flat ifMis totally geodesic.

6. Holomorphic Bisectional Curvature of a GCR-Lightlike Submanifold

Definition 6.1. The holomorphic bisectional for the pair of unit vector fields{X, Y}onMis given byHX, Y gRX, JXJY, Y.

Theorem 6.2. LetMbe a mixed totally geodesicGCR-lightlike submanifold of an indefinite Kaehler manifoldMwithD₀ parallel distribution. ThenHX, Z 0, for any unit vector fieldsX ∈ΓD0 andZ∈ΓM2.

Proof. LetX, Y∈ΓD0andZ∈ΓM2then, by using that hypothesis that the distributionD₀ is a parallel, with respect to∇onM, we havegT∇XZ, Y −g∇XZ, TY gZ,∇XTY 0.

Hence, the non degeneracy of the distributionD0implies that

∇XZ∈Γ D

, 6.1

(12)

for anyZ ∈ ΓM2. Now replacingY byJX, respectively, in 2.15and then taking inner product withJZ, for anyX∈ΓD0andZ∈ΓM2, we get

HX, Z gh^s∇XJX, Z, JZ−g∇Xh^sJX, Z, JZ gh^sJX,∇XZ, JZ g

∇JXh^sX, Z, JZ

−g h^s

∇JXX, Z , JZ

−g h^s

X,∇JXZ , JZ

−g D^s

X, h^lJX, Z , JZ

g D^s

JX, h^lX, Z , JZ

.

6.2

Hence by using thatMis mixed totally geodesic with6.1, the assertion follows.

Theorem 6.3. In order that an indefinite complex space form Mc may admit a mixed totally geodesicGCR-lightlike submanifoldMwith parallel horizontal distributionD₀, it is necessary that c0.

Proof. LetX ∈ΓD0andZ∈ΓM2be unit vector fields, then4.1implies thatHX, Z

−c/2gX, XgZ, Z, then the non degeneracy of the distributions D₀ and M₂ with the Theorem 6.2, we obtainc0. Hence, the result follows.

Lemma 6.4. LetMbe aGCR-lightlike submanifold of an indefinite Kaehler manifoldM. Then we have the following:

iifDdefines a totally geodesic foliation inMthengh^sD, D₀, JD 0,

iiif D₀ is a parallel distribution, with respect to∇, then hZ, JX ChZ, X for any X∈ΓD0, Z∈ΓM2.

Proof. i Let D define a totally geodesic foliation in M this implies that∇XY ∇XY ∈ ΓDandhX, Y 0, for anyX, Y ∈ ΓD. Therefore by using3.10, we obtainA_wYX

−BhX, Y 0. LetZ∈ΓD0, then by using2.8, we get 0gAwYX, Z gh^sX, Z, wY. Thus we havegh^sD, D0, JD 0.

iiLetD₀is a parallel distribution with respect to the induced connection∇, therefore

∇XY ∈ ΓD0, for any Y ∈ ΓD0, X ∈ ΓTM. SinceMis Kaehler manifold, therefore for Z ∈ ΓM2andX ∈ ΓD0, we have∇ZJX J∇ZX. This implies that∇ZJXhZ, JX J∇ZXBhZ, X ChZ, X, then by equating transversal components on both sides, we get the result.

Theorem 6.5. LetMbe aGCR-lightlike submanifold of an indefinite Kaehler manifoldM. IfD₀is parallel with respect to the induced connection∇, andM₂ defines a totally geodesic foliation inM, then

HX, Z ghJX, Z, J∇XZ−g

hX, Z, J∇JXZ g

A_JZJX,∇XZ

−g

AJZX,∇JXZ

2Ch^sX, Z², 6.3

for any unit vector fieldsX ∈ΓD0andZ∈ΓM2.

(13)

Proof. LetX ∈ΓD0andZ∈ΓM2, then the equation of Codazzi2.15becomes g

RX, JXZ, JZ

g∇Xh^sJX, Z, JZ−g

∇JXh^s

X, Z, JZ

g D^s

−g D^s

.

6.4

By using2.13and6.1with theLemma 6.4i, we obtain

HX, Z g

∇^s_JXh^sX, Z, JZ

−g

∇^s_Xh^sJX, Z, JZ g

AJZJX,∇XZ

−g

AJZX,∇JXZ

−g D^s

g D^s

.

6.5

Now by using2.7with theLemma 6.4ii, we have

g

∇^s_Xh^sJX, Z, JZ g

∇Xh^sJX, Z, JZ

−gh^sJX, Z, J∇XZ− Ch^sX, Z²,

6.6

and similarly

g

∇^s_JXh^sX, Z, JZ −g

h^sX, Z, J∇JXZ

Ch^sX, Z². 6.7

By using2.7, we have

g D^s

−g

h^lJX, Z, J∇XZ

, 6.8

g D^s

−g

h^lX, Z, J∇JXZ

. 6.9

Hence by using6.6–6.9in6.5, the result follows.

Definition 6.6see13. AGCR-lightlike submanifoldMof an indefinite Kaehler manifold Mis called aGCR-lightlike product if both the distributionsDandDdefine totally geodesic foliations inM.

Theorem 6.7see14. LetMbe aGCR-lightlike submanifold of an indefinite Kaehler manifoldM.

Then,Mis aGCR-lightlike product if and only if∇XTY 0, for anyX, Y ∈ΓDorX, Y ∈ΓD.

(14)

Theorem 6.8. Let M be a GCR-lightlike submanifold of an indefinite Kaehler manifold M. If

∇XTY 0, for anyX, Y ∈ΓTM, then

HX, Z 2gh^sJX, Z, Jh^sX, Z, 6.10

for any unit vector fieldsX ∈ΓD0andZ∈ΓM2. Proof. Let∇XTY 0, then3.10implies that

AwYXBhX, Y 0, 6.11

for anyX, Y ∈ΓTM. LetY ∈ΓD, X∈ΓTM, then6.11gives

BhX, Y 0. 6.12

LetZ∈ΓDandX ∈ΓD, then using6.11and6.12, we obtain

AwZX0. 6.13

Particularly choosingX ∈ΓD0andZ∈ΓM2in2.15, we get g

RX, JXZ, JZ

g∇Xh^sJX, Z, JZ−g

∇JXh^s

X, Z, JZ g

D^s

−g D^s

. 6.14

Since∇XTY 0 therefore, by usingTheorem 6.7, the distributionsDandDdefine totally geodesic foliations inM. ThenDdefines totally geodesic foliation inMimplies that for any Z₁, Z₂ ∈ΓD, we have∇Z1Z₂ ∈ΓD. Therefore, using3.10and3.11, we getA_wZ₂Z₁ BhZ1, Z₂ 0. By taking inner product withX∈ΓD0and using2.8, we get

gh^sZ1, X, wZ2 0. 6.15

Also,3.11implies thatT∇XZ0, that is,∇XZ∈ΓD. Therefore using2.8,2.13,6.13, and6.15in6.14, we obtain

g

RX, JXZ, JZ g

∇^s_Xh^sJX, Z, JZ

−g

∇^s_JXh^sX, Z, JZ g

D^s

−g D^s

.

6.16

Now using2.6,2.7,2.8, and6.15, we have g

∇^s_Xh^sJX, Z, JZ

gJh^sJX, Z, h^sX, Z. 6.17

(15)

Similarly,

g

∇^s_JXh^sX, Z, JZ

−gJh^sJX, Z, h^sX, Z. 6.18

Also using2.7, we have g

D^s

0, g D^s

0. 6.19

Hence, using6.17–6.19in6.16, the result follows.

Lemma 6.9. Let M be a GCR-lightlike submanifold of an indefinite Kaehler manifold such that

∇XBY 0. Thenh^sJX, Z∈ΓL^⊥₂for anyX∈ΓD0,Z∈ΓM2.

Proof. Since∇XBY 0, therefore, from3.12we haveTAWX 0, for anyX ∈ΓTMand W∈ΓL2, this implies that

AWX ∈Γ D

, 6.20

for anyW ∈ ΓL2andX ∈ ΓTM. Sinceh^sJX, Z ∈ΓSTM^⊥, therefore, to prove that h^sJX, Z ∈ ΓL^⊥₂, it is suﬃcient to prove that gh^sJX, Z, W 0, for any W ∈ ΓL2. Let X ∈ ΓD0 and Z ∈ ΓM2 such that W wZ, we have g∇UZ, X g∇UZ, X g∇UJZ, JX −gAJZU, JX gJAJZU, X, then using2.8we obtain

gh^sJX, Z, W −gJAWZ, X −g∇ZZ, X −gT∇ZZ, TX. 6.21

Since, from3.10, we haveT∇ZZ−AwZZ−BhZ, Z, then using6.20in6.21, the result follows.

Theorem 6.10. LetMbe aGCR-lightlike submanifold of an indefinite Kaehler manifold such that

∇XBY 0, then

HX, Z 2gJh^sX, Z, h^sJX, Z−g

A_JZ∇JXX, Z g

A_JZ∇XJX, Z

, 6.22

Proof. LetX ∈ΓD0andZ∈ΓM2, then, from2.15and2.13, we obtain g

RX, JXZ, JZ g

∇^s_Xh^sJX, Z, JZ

−gh^s∇XJX, Z, JZ

−gh^sJX,∇XZ, JZ−g

∇^s_JXh^sX, Z, JZ g

h^s

∇JXX, Z , JZ

g h^s

X,∇JXZ , JZ g

D^s

−g D^s

,

6.23

(16)

using6.20in2.9, we obtain g

D^s

0, g D^s

0. 6.24

Now consider g

∇^s_Xh^sJX, Z, JZ −g

h^sJX, Z, J∇XZ

gh^sX, Z, Jh^sJX, Z, 6.25

and similarly

g

∇^s_JXh^sX, Z, JZ

−gh^sX, Z, Jh^sJX, Z. 6.26

Also using2.8, we have

gh^s∇XJX, Z, JZ g

AJZ∇XJX, Z , g

h^s

∇JXX, Z , JZ

g

A_JZ∇JXX, Z

. 6.27

Using6.20in2.8, we have

gh^sJX,∇XZ, JZ 0, g h^s

X,∇JXZ , JZ

0. 6.28

Thus using6.24–6.28in6.23, the result follows.

Theorem 6.11. LetMbe a mixed foliateGCR-lightlike submanifold of an indefinite Kaehler manifold M, andSTMis parallel distribution, with respect to the induced connection∇, then

HX, Z 2g

AJZJX, JAJZX

, 6.29

Proof. SinceMis mixed foliate therefore for anyX∈ΓD0andZ∈ΓM2, Codazzi equation 2.15and2.13, imply that

HX, Z gh^s∇XJX, Z, JZ gh^sJX,∇XZ, JZ

−g h^s

∇JXX, Z , JZ

−g h^s

X,∇JXZ , JZ

. 6.30

Since using 2.8 and the hypothesis, we have gAJZX, JW 0 and gAJZX, Jξ gh^sX, Jξ, JZ g∇XJξ, JZ −gξ,∇XZ 0. Therefore, by the definition of a GCR- lightlike submanifold, we have

A_JZX ∈ΓD. 6.31

(17)

Thus, using2.7,2.8, and6.31with the hypothesis, we obtain gh^sJX,∇XZ, JZ g

A_JZJX,∇XZ g

JA_JZJX, J∇XZ g

A_JZJX, JA_JZX

−g

A_JZJX, JD^lX, JZ g

AJZJX, JAJZX .

6.32

Similarly, we obtain

g h^s

X,∇JXZ , JZ

g

AJZX, JAJZJX , g

h^s

∇JXX, Z , JZ

g

AJZZ,∇JXX

−g

∇JXX, D^lJZ, Z , gh^s∇XJX, Z, JZ g

AJZZ,∇XJX

−g

∇XJX, D^lJZ, Z .

6.33

Thus,6.30becomes HX, Z 2g

AJZJX, JAJZX

−g

AJZZ,∇JXX g

∇JXX, D^lJZ, Z g

AJZZ,∇XJX

−g

∇XJX, D^lJZ, Z .

6.34

Since the distributionDis integrable, therefore∇JXX− ∇XJX X, JX X ∈ΓD, then 6.34becomesHX, Z 2gAJZJX, JAJZX−gAJZZ, X gX, D^lJZ, Z. Hence, using the hypothesis and2.8, the result follows.

Acknowledgment

The authors would like to thank the anonymous referee for his/her comments that helped us to improve this paper.

References

1 A. Bejancu, “CR submanifolds of a Kaehler manifold. I,” Proceedings of the American Mathematical Soci- ety, vol. 69, no. 1, pp. 135–142, 1978.

2 A. Bejancu, “CR submanifolds of a Kaehler manifold. II,” Transactions of the American Mathematical Society, vol. 250, pp. 333–345, 1979.

3 A. Bejancu, M. Kon, and K. Yano, “CR-submanifolds of a complex space form,” Journal of Diﬀerential Geometry, vol. 16, no. 1, pp. 137–145, 1981.

4 D. E. Blair and B.-Y. Chen, “On CR-submanifolds of Hermitian manifolds,” Israel Journal of Mathemat- ics, vol. 34, no. 4, pp. 353–363, 1979.

5 B.-Y. Chen, “CR-submanifolds of a Kaehler manifold. I,” Journal of Diﬀerential Geometry, vol. 16, no. 2, pp. 305–322, 1981.

6 K. Yano and M. Kon, “Diﬀerential geometry of CR-submanifolds,” Geometriae Dedicata, vol. 10, no.

1–4, pp. 369–391, 1981.

(18)

7 K. Yano and M. Kon, CR Submanifolds of Kaehlerian and Sasakian Manifolds, vol. 30, Birkh¨auser, Boston, Mass, USA, 1983.

8 R. Penrose, “Physical space-time and nonrealizable CR-structures,” Proceedings of Symposia in Pure Mathematics, vol. 39, pp. 401–422, 1983.

9 K. L. Duggal, “CR-Structures and Lorentzian Geometry,” Acta Applicandae Mathematicae, vol. 7, no. 3, pp. 211–223, 1986.

10 K. L. Duggal, “Lorentzian geometry of CR submanifolds,” Acta Applicandae Mathematicae, vol. 17, no.

2, pp. 171–193, 1989.

11 K. L. Duggal and A. Bejancu, Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications, vol.

364 of Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1996.

12 K. L. Duggal and B. Sahin, “Screen Cauchy Riemann lightlike submanifolds,” Acta Mathematica Hun- garica, vol. 106, no. 1-2, pp. 125–153, 2005.

13 K. L. Duggal and B. Sahin, “Generalized Cauchy-Riemann lightlike submanifolds of Kaehler manifolds,” Acta Mathematica Hungarica, vol. 112, no. 1-2, pp. 107–130, 2006.

14 R. Kumar, S. Kumar, and R. K. Nagaich, “GCR-lightlike product of indefinite Kaehler manifolds,”

ISRN Geometry, vol. 2011, Article ID 531281, 13 pages, 2011.

15 R. S. Kulkarni, “The values of sectional curvature in indefinite metrics,” Commentarii Mathematici Helvetici, vol. 54, no. 1, pp. 173–176, 1979.

16 A. Bonome, R. Castro, E. Garc´ıa-R´ıo, and L. M. Hervella, “On the holomorphic sectional curvature on an indefinite K¨ahler manifold,” Comptes Rendus de l’Acad´emie des Sciences I, vol. 315, no. 11, pp.

1183–1187, 1992.

17 M. Barros and A. Romero, “Indefinite K¨ahler manifolds,” Mathematische Annalen, vol. 261, no. 1, pp.

55–62, 1982.

18 J. K. Beem and P. E. Ehrlich, Global Lorentzian Geometry, vol. 67, Marcel Dekker, New York, NY, USA, 1981.

19 A. Bonome, R. Castro, E. Garc´ıa-R´ıo, L. M. Hervella, and Y. Matsushita, “Null holomorphically flat indefinite almost Hermitian manifolds,” Illinois Journal of Mathematics, vol. 39, no. 4, pp. 635–660, 1995.

20 K. L. Duggal and D. H. Jin, “Totally umbilical lightlike submanifolds,” Kodai Mathematical Journal, vol.

26, no. 1, pp. 49–68, 2003.

(19)

Submit your manuscripts at http://www.hindawi.com

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Mathematics

^{Journal of}

Mathematical Problems in Engineering

Hindawi Publishing Corporation http://www.hindawi.com

Differential Equations

International Journal of

Volume 2014

Journal of

Applied Mathematics

Probability and Statistics

Advances in

Mathematical Physics

Complex Analysis

^{Journal of}

Optimization

^{Journal of}

Combinatorics Operations Research

Journal of Function Spaces

Abstract and Applied Analysis

International Journal of Mathematics and Mathematical Sciences

The Scientific World Journal

Discrete Dynamics in Nature and Society

Decision Sciences

Discrete Mathematics

^{Journal of}

Characterization of Holomorphic Bisectional Curvature of GCR-Lightlike Submanifolds

Research Article