Table of Contents Author Guidelines Submit a Manuscript
International Journal of Mathematics and Mathematical Sciences
Volume 2011 (2011), Article ID 857953, 10 pages
http://dx.doi.org/10.1155/2011/857953
Research Article

On CR-Lightlike Product of an Indefinite Kaehler Manifold

1University College of Engineering, Punjabi University, Patiala, India
2Department of Mathematics, Punjabi University, Patiala, India

Received 20 December 2010; Accepted 3 March 2011

Academic Editor: B. N. Mandal

Copyright © 2011 Rakesh 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 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 [25] 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 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 . 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 -lightlike submanifold and is called the radical distribution on .

Let be a screen distribution which is a semi-Riemannian complementary distribution of in , that is, where 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 Let be a local coordinate neighborhood of and consider the local quasiorthonormal fields of frames of along , on as , where , are local lightlike bases of , and are local orthonormal bases of and , respectively. For this quasi-orthonormal fields of frames, we have the following.

Theorem 2.1 (see [7]). Let be an -lightlike submanifold of a semi-Riemannian manifold . Then, there exists a complementary vector bundle of 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 (2.5), the Gauss and Weingarten formulas are given by 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 operator on and known as shape operator.

According to (2.4), considering the projection morphisms and of on and , respectively, (2.7) and (2.8) give where we put , , ,and .

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 (2.4)-(2.5) and (2.9)–(2.12), we obtain for any , , and .

Let is a projection of on . Now, considering the decomposition (2.3), we can write for any and , where and belong to and , respectively. Here, and are linear connections on and , respectively. By using (2.9)-(2.10) and (2.14), we obtain

Definition 2.2. Let be a real -dimensional indefinite Kaehler manifold and an -dimensional submanifold of . Then, is said to be a CR-lightlike submanifold if the following two conditions are fulfilled:A is distribution on such that Bthere exist vector bundles and over such that where is a nondegenerate distribution on , and are vector sub-bundles of and , respectively.
Clearly, the tangent bundle of a CR-lightlike submanifold is decomposed as where

Theorem 2.3. Let be a 1-lightlike submanifold of codimension 2 of a real 2 m-dimensional indefinite almost Hermitian manifold such that is a distribution on . Then, is a CR-lightlike submanifold.

Proof. Since , therefore, is tangent to . Moreover, and are distributions of rank 1 on , therefore, . This enables one to choose a screen distribution such that it contains . For any , we have and . As is of rank 1, therefore, is also a distribution on such that and , where is a distribution of rank 2 m–5. Now, for any , we have and , therefore, is tangent to and . This implies that has no component in , , and . Thus, lies in and we have where is a nondegenerate distribution; otherwise would be degenerate.
Now, denote , then condition (B) is satisfied, where , , and . Hence the result is present.

3. Mixed Geodesic and CR-Lightlike Product

Definition 3.1. A CR-lightlike submanifold of an indefinite almost Hermitian manifold is called mixed geodesic CR-lightlike submanifolds if the second fundamental form satisfies , for any and .

Definition 3.2. A CR-lightlike submanifold of an indefinite almost Hermitian manifold is called -geodesic (resp., -geodesic) CR-lightlike submanifolds if its second fundamental form , satisfies , for any (resp., ).

Definition 3.3 (see [8]). A CR-lightlike submanifold in a Kaehler manifold is said to be a mixed foliate if the distribution is integrable and is mixed totally geodesic CR-lightlike submanifold.
Now, suppose be a basis of with respect to the basis of , such that is a basis of . Also, consider an orthonormal basis of .

Theorem 3.4. Let be a CR -lightlike submanifold of an indefinite Kaehler manifold . Then, the distribution defines a totally geodesic foliation if is -geodesic.

Proof. Distribution defines a totally geodesic foliation, if and only if,
Since , therefore (3.1) holds, if and only if, , , for all .
Now let , then , similarly . Then, the result follows directly by using hypothesis.

Corollary 3.5. Let be a CR -lightlike submanifold of an indefinite Kaehler manifold . If the distribution defines a totally geodesic foliation, then , for all .

Theorem 3.6 (See [7]). Let be a CR -lightlike submanifold of an indefinite Kaehler manifold . Then, the almost complex distribution is integrable, if and only if, the second fundamental form of satisfies for any .
Now, let and be the projections on and , respectively. Then, one has where and . Clearly, is a tensor field of type and is -valued 1-form on . Clearly, if and only if, . On the other hand, one sets where and are sections of and , respectively.
By using Kaehlerian property of with (2.7) and (2.8), one has the following lemmas.

Lemma 3.7. Let be a CR -lightlike submanifold of an indefinite Kaehler manifold . Then, one has for any , where

Lemma 3.8. Let be a CR -lightlike submanifold of an indefinite Kaehler manifold . Then, one has for any and , where

Theorem 3.9. Let be a CR -lightlike submanifold of an indefinite Kaehler manifold . Then, distribution defines totally geodesic foliation, if and only if, is integrable.

Proof. From (3.6), we obtain for all . Then, taking into account that is symmetric and is torsion free, we obtain which proves the assertion.

Corollary 3.10. If CR -lightlike submanifold in an indefinite Kaehler manifold is mixed foliate, then, using Theorem 3.9 and Corollary 3.5, it is clear that , for any , and , for any .

Theorem 3.11. Let be a CR -lightlike submanifold of an indefinite Kaehler manifold . Then, is mixed geodesic, if and only if, , , for any , .

Proof. Since is a Kaehler manifold, therefore . Since , this implies that , , therefore, . This gives ; comparing transversal parts, we obtain . Thus, is mixed geodesic, if and only if, . This implies .

Lemma 3.12. Let be a CR -lightlike submanifold of an indefinite Kaehler manifold . Then, for any and .

Proof. Let such that . Since is a Kaehler manifold, therefore, , for any . Then, . Comparing the tangential parts, we obtain (3.9).
Let be an indefinite Kaehler manifold of constant holomorphic sectional curvature , then curvature tensor is given by

Theorem 3.13. Let be a mixed foliate CR -lightlike submanifold of an indefinite complex space form . Then, is a complex semi-Euclidean space.

Proof. From (3.16), we have for any and . Since, is a mixed foliate, therefore, for and , we have where and   . By hypothesis, is mixed geodesic, therefore,   . Again, by using hypothesis and Theorem 3.9, the distribution defines totally geodesic foliation. Hence, . Therefore, we have ; using Lemma 3.12 here, we obtain .
By using, is mixed geodesic and Corollary 3.10, we obtain Therefore, from (3.17)–(3.19), we get . Since and are non-degenerate, therefore, we have , which completes the proof.

Definition 3.14 (see [7]). A CR-lightlike submanifold of a Kaehler manifold is called a CR-lightlike product if both the distribution and define totally geodesic foliations on .

Theorem 3.15 (see [7]). Let be a CR -lightlike submanifold of an indefinite Kaehler manifold . Then, defines a totally geodesic foliation on , if and only if, for any , has no component in .

Theorem 3.16 (see [7]). Let be a CR -lightlike submanifold of an indefinite Kaehler manifold . Then, defines a totally geodesic foliation on , if and only if, for any , one has

Theorem 3.17 (see [3]). A CR -submanifold of a Kaehler manifold is a CR -product, if and only if, is parallel, that is, , where .

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 is a CR -lightlike product, if and only if, is parallel, that is, .

Proof. If is parallel, then (3.5), gives for any . In particular, if , then . Hence, (3.23) implies , thus, Theorem 3.15 follows.
Let , then , then (3.23) implies , and, hence, (3.22) follows. For any , . Thus, (3.21) holds well. Since for any , let , then we have . Hence, . Therefore, , for any and and consequently . Hence, (3.20) holds well. Thus, is a CR-lightlike product in .
Conversely, if is a CR-lightlike product, then by definition, and both defines totally geodesic foliation, that is, , for any and , for any . If for any , then by comparing the transversal parts in the simplification of Kaehlerian property of , we get Therefore, from (2.7), (3.7), and (3.24), we may prove that or .
Let , for any , then by comparing the tangential parts in the simplification of Kaehlerian property of , we get Thus, from (3.5) and (3.25), we obtain or . Hence, the proof is complete.

Theorem 3.19. Let be a CR -lightlike submanifold of an indefinite Kaehler manifold . If is a CR -lightlike product, then it is a mixed foliate CR-lightlike submanifold.

Proof. Since is a CR-lightlike product then by Theorem 3.9, it is clear that the distribution is integrable. Also, by using Theorem 3.18, for CR-lightlike product, , then (3.5) gives for any . Then, in particular, if , then . Hence, (3.26) implies , that is, for any and .
Next, by the definition of CR-lightlike submanifold is mixed geodesic, if and only if, and , for any , , and . By hypothesis, we have for any ; let and , then we have . Hence, . Therefore, , since , therefore, there exists such that ; this gives , by virtue of (3.27).
Similarly, . Now, from (3.10) and (3.12), we have ; since , therefore , so we have . Therefore, by using (3.27), we have ; this gives , for all . Hence, the proof is complete.

References

  1. 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
  2. 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
  3. 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
  4. B.-Y. Chen, “CR-submanifolds of a Kaehler manifold. II,” Journal of Differential Geometry, vol. 16, no. 3, pp. 493–509, 1981. View at Google Scholar · View at Zentralblatt MATH
  5. K. Yano and M. Kon, “CR submanifolds of a complex projective space,” Journal of Differential Geometry, vol. 16, no. 3, pp. 431–444, 1981. View at Google Scholar · View at Zentralblatt MATH
  6. D. N. Kupeli, Singular Semi-Riemannian Geometry, vol. 366 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1996.
  7. 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.
  8. R. Sharma and K. L. Duggal, “Mixed foliate CR-submanifolds of indefinite complex space-forms,” Annali di Matematica Pura ed Applicata. Serie Quarta, vol. 149, pp. 103–111, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH