`ISRN GeometryVolumeΒ 2011Β (2011), Article IDΒ 161523, 9 pageshttp://dx.doi.org/10.5402/2011/161523`
Research Article

## Some Results on Lorentzian Para-Sasakian Manifolds

Department of Mathematics, Kuvempu University, Shankaraghatta, Karnataka, Shimoga 577 451, India

Received 3 June 2011; Accepted 3 July 2011

Copyright Β© 2011 Venkatesha 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

The object of the present paper is to study Lorentzian para-Sasakian (briefly LP-Sasakian) manifolds satisfying certain conditions on the -curvature tensor.

#### 1. Introduction

In 1989, Matsumoto [1] introduced the notion of Lorentzian para-Sasakian manifold. Then, Mihai and Roşca [2] introduced the same notion independently and they obtained several results on this manifold. LP-Sasakian manifolds have also been studied by Matsumoto and Mihai [3], Mihai et al. [4], and Venkatesha and Bagewadi [5].

On the other hand, Pokhariyal and Mishra [6] have introduced new curvature tensor called -curvature tensor in a Riemannian manifold and studied their properties. Further, Pokhariyal [7] has studied some properties of this curvature tensor in a Sasakian manifold. Matsumoto et al. [8], and Yìldìz and De [9] have studied -curvature tensor in P-Sasakian and Kenmotsu manifolds, respectively.

In the present paper, we study some curvature conditions on LP-Sasakian manifolds. Firstly, we study LP-Sasakian manifolds satisfying and -semisymmetric LP-Sasakian manifolds. Further, we study LP-Sasakian manifolds which satisfy , and , where is the projective curvature tensor, is the -projective curvature tensor, and is the conformal curvature tensor.

#### 2. Preliminaries

An -dimensional differentiable manifold is called an LP-Sasakian manifold [1, 2] if it admits a tensor field , a contravariant vector field , a 1-form , and a Lorentzian metric which satisfy where denotes the operator of covariant differentiation with respect to the Lorentzian metric .

It can be easily seen that, in an LP-Sasakian manifold, the following relations hold: If we put for any vector fields and , then the tensor field is a symmetric (0, 2) tensor field [1]. Also, since the 1-form is closed in an LP-Sasakian manifold, we have [1, 4] for any vector fields and .

Also in an LP-Sasakian manifold, the following relations hold [3, 4]: for any vector fields , and , where is the Riemannian curvature tensor and is the Ricci tensor of .

An LP-Sasakian manifold is said to be Einstein if its Ricci tensor is of the form for any vector fields and , where is a function on .

In [6], Pokhariyal and Mishra have defined the curvature tensor , given by where is a Ricci tensor of type (0, 2).

Now, consider an LP-Sasakian manifold satisfying ; then, (2.11) becomes Taking in (2.12) and using (2.7) and (2.8), we have Therefore, is an Einstein manifold.

Again using (2.13) in (2.12), we get

Corollary 2.1. An LP-Sasakian manifold satisfying is a space of constant curvature −1, that is, it is locally isometric to the hyperbolic space.

Definition 2.2. An LP-Sasakian manifold is called -semisymmetric if it satisfies where is to be considered as a derivation of the tensor algebra at each point of the manifold for tangent vectors and .
It can be easily shown that in an LP-Sasakian manifold the -curvature tensor satisfies the condition

Theorem 2.3. A -semisymmetric LP-Sasakian manifold is an Einstein manifold.

Proof. Since , we have Putting in (2.17) and then taking the inner product with , we obtain Using (2.6) in (2.18), we obtain By using (2.16) in (2.19), we get In view of (2.11) and (2.20), it follows that Contracting (2.21), we have Therefore, is an Einstein manifold.

Again using (2.22) in (2.12), we get

Corollary 2.4. A -semisymmetric LP-Sasakian manifold is a space of constant curvature −1, that is, it is locally isometric to the hyperbolic space.

#### 3. LP-Sasakian Manifolds Satisfying π ( π , π ) β π 2 = 0

The projective curvature tensor is defined as [10] Using (2.6) and (2.8), (3.1) reduces to Let us suppose that in an LP-Sasakian manifold This condition implies that Putting in (3.4) and then taking the inner product with , we obtain Using (3.2) in (3.5), we obtain By using (2.16) in (3.6), we get Taking in (3.7) and using (2.11) and (2.6), we have This implies that From this, we get Thus, we can state the following.

Theorem 3.1. An LP-Sasakian manifold satisfying the condition is an Einstein manifold.

#### 4. LP-Sasakian Manifold Satisfying ξ π ( π , π ) β π 2 = 0

The -projective curvature tensor is defined as [11] Using (2.6) and (2.8), (4.1) reduces to Suppose that in an LP-Sasakian manifold This condition implies that Putting in (4.4) and then taking the inner product with , we obtain Using (4.2) in (4.5), we obtain By using (2.16) in (4.6), we get Taking in (4.7) and using (2.11) and (2.6), we have This implies that which gives Thus, we can state the following.

Theorem 4.1. An LP-Sasakian manifold satisfying the condition is an Einstein manifold.

#### 5. LP-Sasakian Manifolds Satisfying πΆ ( π , π ) β π 2 = 0

The conformal curvature tensor is defined as [12] Using (2.6) and (2.8), (5.1) reduces to Now consider an LP-Sasakian manifold satisfying This condition implies that Putting in (5.4) and then taking the inner product with , we obtain Using (5.2) in (5.5), we obtain By using (2.16) in (5.6), we get Taking in (5.7) and then using (2.11) and (2.6), we have This implies thatand it follows that Thus, we can state the following.

Theorem 5.1. An LP-Sasakian manifold satisfying the condition is an Einstein manifold.

#### Acknowledgment

The authors express their thanks to DST (Department of Science and Technology), Government of India for providing financial assistance under major research project.

#### References

1. K. Matsumoto, βOn Lorentzian paracontact manifolds,β Bulletin of the Yamagata University. Natural Science, vol. 12, no. 2, pp. 151β156, 1989.
2. I. Mihai and R. Roşca, βOn Lorentzian $P$-Sasakian manifolds,β in Classical Analysis (Kazimierz Dolny, 1991), pp. 155β169, World Scientific, River Edge, NJ, USA, 1992.
3. K. Matsumoto and I. Mihai, βOn a certain transformation in a Lorentzian para-Sasakian manifold,β Tensor, vol. 47, no. 2, pp. 189β197, 1988.
4. I. Mihai, A. A. Shaikh, and U. C. De, βOn Lorentzian para-Sasakian manifolds,β Rendiconti del Seminario Matematico di Messina, no. 3, pp. 149β158, 1999.
5. Venkatesha and C. S. Bagewadi, βOn concircular $\varphi$-recurrent LP-Sasakian manifolds,β Differential Geometry—Dynamical Systems, vol. 10, pp. 312β319, 2008.
6. G. P. Pokhariyal and R. S. Mishra, βCurvature tensors' and their relativistics significance,β Yokohama Mathematical Journal, vol. 18, pp. 105β108, 1970.
7. G. P. Pokhariyal, βStudy of a new curvature tensor in a Sasakian manifold,β Tensor, vol. 36, no. 2, pp. 222β226, 1982.
8. K. Matsumoto, S. Ianuş, and I. Mihai, βOn $P$-Sasakian manifolds which admit certain tensor fields,β Publicationes Mathematicae Debrecen, vol. 33, no. 3-4, pp. 199β204, 1986.
9. A. Yíldíz and U. C. De, βOn a type of Kenmotsu manifolds,β Differential Geometry—Dynamical Systems, vol. 12, pp. 289β298, 2010.
10. K. Yano and M. Kon, Structures on Manifolds, vol. 3 of Series in Pure Mathematics, World Scientific, Singapore, 1984.
11. G. P. Pokhariyal and R. S. Mishra, βCurvature tensors and their relativistic significance. II,β Yokohama Mathematical Journal, vol. 19, no. 2, pp. 97β103, 1971.
12. L. P. Eisenhart, Riemannian Geometry, Princeton University Press, Princeton, NJ, USA, 1949.