Advances in Mathematical Physics

Volume 2012 (2012), Article ID 356263, 18 pages

http://dx.doi.org/10.1155/2012/356263

## Characterization of Holomorphic Bisectional Curvature of *GCR*-Lightlike Submanifolds

^{1}Department of Applied Sciences, Rayat Institute of Engineering & Information Technology, Railmajra, SBS Nagar, Punjab 144533, India^{2}Department of Basic and Applied Sciences, University College of Engineering, Punjabi University, Patiala 147002, India^{3}Department of Mathematics, Punjabi University, Patiala 147002, India

Received 27 March 2012; Accepted 19 May 2012

Academic Editor: M. Lakshmanan

Copyright © 2012 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.

#### Abstract

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 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 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 manifold. Duggal [9, 10] studied the geometry of 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 structures in relativity and growing importance of lightlike submanifolds in mathematical physics and relativity, Duggal and Bejancu [11] introduced the notion of -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 -non degenerate submanifolds, -lightlike submanifolds do not include invariant (complex) and totally real lightlike submanifolds. Therefore, Duggal and Sahin [12] introduced -lightlike submanifolds of indefinite Kaehler manifold which contain complex and totally real subcases but there was no inclusion relation between and cases. Later on, Duggal and Sahin [13] introduced -lightlike submanifolds of indefinite Kaehler manifolds, which behaves as an umbrella of invariant (complex), screen real and -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 -lightlike submanifolds of indefinite Kaehler manifolds and obtained some characterization theorems for a -lightlike submanifold to be a -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 -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 -lightlike submanifolds of an indefinite complex space form. In Section 5, we established a condition for a -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 be a real -dimensional semi-Riemannian manifold of constant index such that , , an -dimensional submanifold of and the induced metric of on . If is degenerate on the tangent bundle of , then is called a lightlike submanifold of (see [11]). For a degenerate metric on , is a degenerate -dimensional subspace of . Thus both and are degenerate orthogonal subspaces but no longer complementary. In this case, there exists a subspace which is known as radical (null) subspace. If the mapping defines a smooth distribution on of rank , then the submanifold of is called an -lightlike submanifold and is called the radical distribution on .

Screen distribution is a semi-Riemannian complementary distribution of in , that is and is a complementary vector subbundle to in . Let and be complementary (but not orthogonal) vector bundles to in and to in , respectively. Then we have For a quasi-orthonormal fields of frames on , we have the following.

Theorem 2.1 (see [11]). * Let be an -lightlike submanifold of a semi-Riemannian manifold . Then there exists a complementary vector bundle of Rad in and a basis of consisting of smooth section of , where is a coordinate neighborhood of such that
**
where is a lightlike basis of . *

Let be the Levi-Civita connection on , then, according to the decomposition (2.3), the Gauss and Weingarten formulas are given by the following: for any and , where and belong to and , respectively. Here is a torsion-free linear connection on , is a symmetric bilinear form on which is called second fundamental form, and is a linear a operator on and known as shape operator.

According to (2.2) considering the projection morphisms and of on and , respectively, then Gauss and Weingarten formulas become where we put , . As and are valued and valued, respectively, therefore they are called the lightlike second fundamental form and the screen second fundamental form on . In particular, where , and . Using (2.6) and (2.7), we obtain for any , , and .

Let be the projection morphism of on , then, using (2.1), we can induce some new geometric objects on the screen distribution on as follows: for any and , where and , belong to and , respectively. Using (2.6) and (2.10), we obtain for any and .

Denote by and the curvature tensors of and , respectively, then by straightforward calculations ([11]), we have where Then Codazzi equation is given, respectively, by the following: Barros and Romero [17] defined indefinite Kaehler manifolds as follows.

*Definition 2.2. *Let be an indefinite almost Hermitian manifold and the Levi-Civita connection on , with respect to an indefinite metric . Then is called an indefinite Kaehler manifold if is parallel, with respect to , that is

#### 3. Generalized Cauchy-Riemann Lightlike Submanifolds

*Definition 3.1. * Let be a real lightlike submanifold of an indefinite Kaehler manifold , then is called a generalized Cauchy-Riemann -lightlike submanifold if the following conditions are satisfied.(A) There exist two subbundles and of such that
(B) There exist two subbundles and of such that
where is a nondegenerate distribution on , and are vector bundle of and , respectively.

Then the tangent bundle of is decomposed as , where . M is called a proper -lightlike submanifold if , and . Let and be the projections on , , and , respectively. Then, for any , we have applying to (3.3), we obtain and we can write (3.4) as follwos: where and are the tangential and transversal components of , respectively. Similarly, for any , where and are the sections of and , respectively. Applying to (3.5) and (3.6), we get

Differentiating (3.4) and using (2.6), (2.7), and (3.6), we have Using Kaehlerian property of with (2.7), we have the following lemmas.

Lemma 3.2. *Let be a -lightlike submanifold of an indefinite Kaehlerian manifold . Then one has
**
where and
*

Lemma 3.3. *Let be a -lightlike submanifold of an indefinite Kaehlerian manifold . Then one has
**
where , , and
*

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

Let be a complex space form of constant holomorphic curvature . Then the curvature tensor of is given by the following: for vector fields on . Using (4.1) and (2.12), we obtain Using (2.8) in (4.2), we obtain Then the sectional curvature of determined by orthonormal vectors and of and given by the following:

Corollary 4.1. *Let be a -lightlike submanifold of an indefinite Complex space form . Then sectional curvature of is given by , if *(i)* defines a totally geodesic foliation in , *(ii)* defines a totally geodesic foliation in , *(iii)* is totally geodesic in . *

*Definition 4.2. *The holomorphic sectional curvature of determined by a unit vector is the sectional curvature of a plane section .

Then using (2.11) and (4.4), for a unit vector field , we get From (3.8), for any , we have and further using (3.7) and (4.6), we have Hence using (4.6) and (4.7) in (4.5), we obtain the expression for holomorphic sectional curvature as follows:

Theorem 4.3. *Let be a -lightlike submanifold of an indefinite complex space form . If is totally geodesic in , then , for any unit vector field . *

* Proof. *Using the hypothesis in (4.8), we get . Hence the result follows.

Theorem 4.4 4.4 (see [13]). *Let be a -lightlike submanifold of an indefinite Kaehler manifold , then the distribution is integrable if and only if , for any . *

Theorem 4.5. *Let be a -lightlike submanifold of an indefinite complex space form , and is integrable, then for any unit vector field . *

* Proof. *Since is integrable therefore using Theorem 4.4, we have , for any unit vector field . Therefore, from (4.5), we obtain

Theorem 4.6. *A -lightlike submanifold of an indefinite complex space form is -totally geodesic if and only if *(i)* is integrable, *(ii)*, for any unit vector field . *

*Proof. *Proof follows from (4.9).

Theorem 4.7 4.7 (see [13]). *Let be a -lightlike submanifold of an indefinite Kaehler manifold . Then the distribution defines a totally geodesic foliation in if and only if , for any . *

Theorem 4.8. * Let be a -lightlike submanifold of an indefinite complex space form , and defines a totally geodesic foliation in , then , for any unit vector field . *

*Proof. *Since defines a totally geodesic foliation in , therefore by definition , this implies that . Also by using Theorem 4.7, we have , for any ; hence, (4.8) becomes and the result follows.

*Definition 4.9. *The horizontal distribution is called parallel with respect to the induced connection on if for any and .

Theorem 4.10. *Let be a -lightlike submanifold of an indefinite complex space form and is parallel, with respect to , then , for any unit vector field . *

* Proof. *Since is parallel, with respect to the induced connection on , therefore and , for any and . Hence, from (4.8), we obtain , and then by using (3.6), we get . Hence the result is complete.

Lemma 4.11. * Let be a -lightlike submanifold of an indefinite Kaehler manifold . If the distribution defines a totally geodesic foliation in , then is -geodesic. *

* Proof. * By the definition of -lightlike submanifold, is -geodesic if , for any , , and . Since defines a totally geodesic foliation in , therefore and . Hence, the assertion follows.

Theorem 4.12. *Let be a -lightlike submanifold of an indefinite complex space form . If defines a totally geodesic foliation in , then , for any unit vector field . *

* Proof. * The result follows directly using Lemma 4.11 and (4.8).

#### 5. Null Holomorphically Flat -Lightlike Submanifold

Let and be a null vector of . A plane of is called a null plane directed by if it contains , , for any , and there exists such that . Following Beem-Ehrlich [18], the null sectional curvature of , with respect to and , as a real number, is defined as follows: where is an arbitrary non null vector in . Clearly is independent of but depends in a quadratic fashion on .

Consider and a null plane of directed by , then the null sectional curvature of , with respect to and , as a real number is defined as where is an arbitrary non-null vector in .

Let be a -lightlike submanifold of an indefinite complex space form then using (4.3), the null sectional curvature of , with respect to , is given by the following: Then, using (2.11), we obtain

We know that a plane is called holomorphic if it remains invariant under the action of the almost complex structure , that is, if . The sectional curvature associated with the holomorphic plane is called the holomorphic sectional curvature, denoted by and given by . The holomorphic plane is called null or degenerate if and only if is a null vector. A manifold is called null holomorphically flat if the curvature tensor satisfies (see [19]). for all null vectors . Put , then, from (5.4), we obtain

*Definition 5.1 (see [20]). *A lightlike submanifold of a semi-Riemannian manifold is said to be a totally umbilical in if there is a smooth transversal vector field on , called the transversal curvature vector field of , such that, for ,
Using (2.6), it is clear that is a totally umbilical if and only if on each coordinate neighborhood there exist smooth vector fields and such that
for and . A lightlike submanifold is said to be totally geodesic if , for any . Therefore, in other words, a lightlike submanifold is totally geodesic if and .

Let be a totally umbilical lightlike submanifold, then, using above definition, we have and , for any . Thus, from (5.6), we have the following theorem.

Theorem 5.2. *Let be a -lightlike submanifold of an indefinite complex space form . If is totally umbilical lightlike submanifold in , then is null holomorphically flat. *

Moreover, from (5.6), it is clear that the expression of is expressed in terms of screen second fundamental forms of , thus -lightlike submanifold of an indefinite complex space form is null holomorphically flat if is totally geodesic.

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

*Definition 6.1. *The holomorphic bisectional for the pair of unit vector fields on is given by .

Theorem 6.2. *Let be a mixed totally geodesic -lightlike submanifold of an indefinite Kaehler manifold with parallel distribution. Then , for any unit vector fields and . *

* Proof. *Let and then, by using that hypothesis that the distribution is a parallel, with respect to on , we have . Hence, the non degeneracy of the distribution implies that

for any . Now replacing by , respectively, in (2.15) and then taking inner product with , for any and , we get
Hence by using that is mixed totally geodesic with (6.1), the assertion follows.

Theorem 6.3. *In order that an indefinite complex space form may admit a mixed totally geodesic -lightlike submanifold with parallel horizontal distribution , it is necessary that .*

* Proof. *Let and be unit vector fields, then (4.1) implies that , then the non degeneracy of the distributions and with the Theorem 6.2, we obtain . Hence, the result follows.

Lemma 6.4. * Let be a -lightlike submanifold of an indefinite Kaehler manifold . Then we have the following:*(i)*if defines a totally geodesic foliation in then ,*(ii)*if is a parallel distribution, with respect to , then for any .*

*Proof. *(i) Let define a totally geodesic foliation in this implies that and , for any . Therefore by using (3.10), we obtain . Let , then by using (2.8), we get . Thus we have .

(ii) Let is a parallel distribution with respect to the induced connection , therefore , for any . Since is Kaehler manifold, therefore for and , we have . This implies that , then by equating transversal components on both sides, we get the result.

Theorem 6.5. *Let be a -lightlike submanifold of an indefinite Kaehler manifold . If is parallel with respect to the induced connection , and defines a totally geodesic foliation in , then
**
for any unit vector fields and . *

* Proof. * Let and , then the equation of Codazzi (2.15) becomes
By using (2.13) and (6.1) with the Lemma 6.4 (i), we obtain
Now by using (2.7) with the Lemma 6.4 (ii), we have
and similarly
By using (2.7), we have
Hence by using (6.6)–(6.9) in (6.5), the result follows.

*Definition 6.6 6.6 (see [13]). *A -lightlike submanifold of an indefinite Kaehler manifold is called a -lightlike product if both the distributions and define totally geodesic foliations in .

Theorem 6.7 (see [14]). *Let be a -lightlike submanifold of an indefinite Kaehler manifold . Then, is a -lightlike product if and only if , for any or .*

Theorem 6.8. * Let be a -lightlike submanifold of an indefinite Kaehler manifold . If , for any , then
**
for any unit vector fields and . *

* Proof. *Let , then (3.10) implies that
for any . Let , then (6.11) gives
Let and , then using (6.11) and (6.12), we obtain
Particularly choosing and in (2.15), we get
Since therefore, by using Theorem 6.7, the distributions and define totally geodesic foliations in . Then defines totally geodesic foliation in implies that for any , we have . Therefore, using (3.10) and (3.11), we get . By taking inner product with and using (2.8), we get
Also, (3.11) implies that , that is, . Therefore using (2.8), (2.13), (6.13), and (6.15) in (6.14), we obtain
Now using (2.6), (2.7), (2.8), and (6.15), we have
Similarly,
Also using (2.7), we have
Hence, using (6.17)–(6.19) in (6.16), the result follows.

Lemma 6.9. *Let be a -lightlike submanifold of an indefinite Kaehler manifold such that . Then for any , . *

* Proof. *Since , therefore, from (3.12) we have , for any and , this implies that
for any and . Since , therefore, to prove that , it is sufficient to prove that , for any . Let and such that , we have , then using (2.8) we obtain
Since, from (3.10), we have , then using (6.20) in (6.21), the result follows.

Theorem 6.10. *Let be a -lightlike submanifold of an indefinite Kaehler manifold such that , then
**
for any unit vector fields and . *

* Proof. *Let and , then, from (2.15) and (2.13), we obtain
using (6.20) in (2.9), we obtain
Now consider
and similarly
Also using (2.8), we have
Using (6.20) in (2.8), we have
Thus using (6.24)–(6.28) in (6.23), the result follows.

Theorem 6.11. *Let be a mixed foliate -lightlike submanifold of an indefinite Kaehler manifold , and is parallel distribution, with respect to the induced connection , then
**
for any unit vector fields and . *

*Proof. *Since is mixed foliate therefore for any and , Codazzi equation (2.15) and (2.13), imply that
Since using (2.8) and the hypothesis, we have and . Therefore, by the definition of a -lightlike submanifold, we have
Thus, using (2.7), (2.8), and (6.31) with the hypothesis, we obtain
Similarly, we obtain
Thus, (6.30) becomes
Since the distribution is integrable, therefore , then (6.34) becomes . Hence, using the hypothesis and (2.8), 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

- A. Bejancu, “CR submanifolds of a Kaehler manifold. I,”
*Proceedings of the American Mathematical Society*, vol. 69, no. 1, pp. 135–142, 1978. View at Google Scholar · View at Zentralblatt MATH - A. Bejancu, “CR submanifolds of a Kaehler manifold. II,”
*Transactions of the American Mathematical Society*, vol. 250, pp. 333–345, 1979. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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. View at Google Scholar · View at Zentralblatt MATH - D. E. Blair and B.-Y. Chen, “On CR-submanifolds of Hermitian manifolds,”
*Israel Journal of Mathematics*, vol. 34, no. 4, pp. 353–363, 1979. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - B.-Y. Chen, “CR-submanifolds of a Kaehler manifold. I,”
*Journal of Differential Geometry*, vol. 16, no. 2, pp. 305–322, 1981. View at Google Scholar · View at Zentralblatt MATH - K. Yano and M. Kon, “Differential geometry of CR-submanifolds,”
*Geometriae Dedicata*, vol. 10, no. 1–4, pp. 369–391, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - K. Yano and M. Kon,
*CR Submanifolds of Kaehlerian and Sasakian Manifolds*, vol. 30, Birkhäuser, Boston, Mass, USA, 1983. - R. Penrose, “Physical space-time and nonrealizable CR-structures,”
*Proceedings of Symposia in Pure Mathematics*, vol. 39, pp. 401–422, 1983. View at Google Scholar - K. L. Duggal, “CR-Structures and Lorentzian Geometry,”
*Acta Applicandae Mathematicae*, vol. 7, no. 3, pp. 211–223, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - K. L. Duggal, “Lorentzian geometry of CR submanifolds,”
*Acta Applicandae Mathematicae*, vol. 17, no. 2, pp. 171–193, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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. - K. L. Duggal and B. Sahin, “Screen Cauchy Riemann lightlike submanifolds,”
*Acta Mathematica Hungarica*, vol. 106, no. 1-2, pp. 125–153, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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. View at Publisher · View at Google Scholar - R. S. Kulkarni, “The values of sectional curvature in indefinite metrics,”
*Commentarii Mathematici Helvetici*, vol. 54, no. 1, pp. 173–176, 1979. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. Bonome, R. Castro, E. García-Río, and L. M. Hervella, “On the holomorphic sectional curvature on an indefinite Kähler manifold,”
*Comptes Rendus de l'Académie des Sciences I*, vol. 315, no. 11, pp. 1183–1187, 1992. View at Google Scholar · View at Zentralblatt MATH - M. Barros and A. Romero, “Indefinite Kähler manifolds,”
*Mathematische Annalen*, vol. 261, no. 1, pp. 55–62, 1982. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. K. Beem and P. E. Ehrlich,
*Global Lorentzian Geometry*, vol. 67, Marcel Dekker, New York, NY, USA, 1981. - A. Bonome, R. Castro, E. García-Río, 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. View at Google Scholar · View at Zentralblatt MATH - K. L. Duggal and D. H. Jin, “Totally umbilical lightlike submanifolds,”
*Kodai Mathematical Journal*, vol. 26, no. 1, pp. 49–68, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH