Some Results on Lorentzian Para-Sasakian Manifolds
Venkatesha,1C. S. Bagewadi,1and K. T. Pradeep Kumar1
Academic Editor: M. Dunajski
Received03 Jun 2011
Accepted03 Jul 2011
Published15 Aug 2011
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 RoΕca [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.
Corollary 2.1. An LP-Sasakian manifold satisfying is a space of constant curvature β1, 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.
Corollary 2.4. A -semisymmetric LP-Sasakian manifold is a space of constant curvature β1, that is, it is locally isometric to the hyperbolic space.
3. LP-Sasakian Manifolds Satisfying
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
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
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.
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