ISRN Geometry

Volume 2012 (2012), Article ID 970682, 18 pages

http://dx.doi.org/10.5402/2012/970682

## Curvature Properties and -Einstein -Contact Metric Manifolds

Department of Mathematics, Bangalore University, Central College Campus, Bangalore 560 001, India

Received 16 September 2012; Accepted 2 October 2012

Academic Editors: G. Martin, C. Qu, and A. Viña

Copyright © 2012 H. G. Nagaraja and C. R. Premalatha. 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 study curvature properties in -contact metric manifolds. We give the characterization of -Einstein -contact metric manifolds with associated scalars.

#### 1. Introduction

The class of -contact manifolds [1] is of interest as it contains both the classes of Sasakian and non-Sasakian cases. The contact metric manifolds for which the characteristic vector field belongs to -nullity distribution are called contact metric manifolds. Boeckx [2] gave a classification of -contact metric manifolds. Sharma [3], Papantoniou [4], and many others have made an investigation of -contact metric manifolds. A special class of -contact metric manifolds called -contact metric manifolds was studied by authors [5, 6] and others. In this paper we study -contact metric manifolds by considering different curvature tensors on it (Table 1). We characterize -Einstein -contact metric manifolds with associated scalars by considering symmetry, -symmetry, semisymmetry, -recurrent, and flat conditions on -contact metric manifolds. The paper is organized as follows: In Section 2, we give some definitions and basic results. In Section 3, we consider conharmonically symmetric, conharmonically semisymmetric, -conharmonically flat, -conharmonically flat, and -recurrent -contact metric manifolds and we prove that such manifolds are -Einstein or -parallel or cosymplectic depending on the conditions. In Section 4, we prove that -conformally flat -contact metric manifold reduces to -contact metric manifold if and only if it is an -Einstein manifold. Further we prove conformally Ricci-symmetric and -conformally flat -contact metric manifolds are -Einstein. In Section 5, we prove that pseudoprojectively symmetric and pseudoprojectively Ricci-symmetric -contact metric manifolds are -Einstein. In Section 6 we consider Ricci-semisymmetric -contact metric manifolds and prove that such manifolds are -Einstein. In all the cases where -contact metric manifold is an -Einstein manifold, we obtain associated scalars in terms of and .

#### 2. Preliminaries

A dimensional -differentiable manifold is said to admit an almost contact metric structure if it satisfies the following relations [7, 8] where is a tensor field of type (1,1), is a vector field, is a 1-form, and is a Riemannian metric on . A manifold equipped with an almost contact metric structure is called an almost contact metric manifold. An almost contact metric manifold is called a contact metric manifold if it satisfies for all vector fields , .

The (1,1) tensor field defined by , where denotes the Lie differentiation, is a symmetric operator and satisfies , , and . Further we have [1] where denotes the Riemannian connection of .

The -nullity distribution of a contact metric manifold is a distribution [1] for any vector fields and on .

*Definition 2.1. *A contact metric manifold is said to be (i)Einstein if , where is a constant and is the Ricci tensor, (ii)-Einstein if , where and are smooth functions.

A contact metric manifold with is called a -contact metric manifold. In a -contact metric manifold, we have If , , then the manifold becomes Sasakian [1], and if , then the notion of -nullity distribution reduces to -nullity distribution [9]. If , then -contact metric manifold is locally isometric to the product . In a (2n+1)-dimensional -contact metric manifold, we have the following [1]: where is the Ricci operator and is the scalar curvature of .

Throughout this paper denotes (2n+1)-dimensional -contact metric manifold.

#### 3. Conharmonic Curvature Tensor in -Contact Metric Manifolds

The conharmonic curvature tensor in is given by [10] A -contact metric manifold is said to be (1)conharmonically symmetric if (2)conharmonically semisymmetric if

##### 3.1. Conharmonically Symmetric -Contact Metric Manifolds

Differentiating (3.1) covariantly with respect to , we obtain If is conharmonically symmetric, then, from (3.4), we obtain Differentiating (2.6) covariantly with respect to and using (2.4), we obtain Differentiating (2.10) covariantly with respect to and using (2.11), (2.4), we have where From (3.7), we obtain Taking in (3.5) and using (3.6), (3.7), and (3.9), we obtain where Contracting (3.10) with and using (2.1), we obtain From (3.12), we get either or Taking in (3.13) and using (2.1), we obtain Taking in (3.14), we obtain Since , from (3.15), it follows that if and only if

In view of (2.4), the above equation gives that reduces to a cosymplectic manifold. Thus we have is cosymplectic if and only if .

Further from (2.10) and Definition 2.1, we have is -Einstein with if and only if . Thus we have the following.

Theorem 3.1. *In a conharmonically symmetric -contact metric manifold , the following statements are equivalent. *(1)* is cosymplectic. *(2)* is -Einstein with . *(3)*. *

##### 3.2. Conharmonically Semisymmetric -Contact Metric Manifolds

Suppose . Then from (3.3), we have Using (2.5), (2.9), (2.6), (2.10), and (2.11) in (3.17) and taking , we get Taking and using (2.9), (2.6), and (2.11), we obtain That is, is an -Einstein manifold.

Conversely, suppose in the relation (3.19) holds. Then we have Using (3.19) in (3.20), we get which implies that is conharmonically semisymmetric. Thus we have the following.

Theorem 3.2. *A -contact metric manifold is conharmonically semisymmetric if and only if it is -Einstein with and .*

##### 3.3. -Conharmonically Flat -Contact Metric Manifolds

Suppose is -conharmonically flat, that is, for all vector fields , , , . Then from (3.1), we obtain Let be a local orthonormal basis of the tangent space at each in . Then in , the following relations hold: Taking in (3.21) and summing up from 1 to , we have Using (2.13), (3.22), in (3.24), we obtain Replacing by and by in (3.25) and using (2.1), we have Taking in (3.26) and taking summation over to , we obtain .

Substituting this in (3.26) and taking the covariant derivative with respect to , we obtain That is, is -parallel.

Further substituting in (2.12), we obtain Thus from the above discussions we can state the following.

Theorem 3.3. *In a -dimensional -conharmonically flat -contact metric manifold, Ricci tensor is -parallel and .*

##### 3.4. -Conharmonically Flat -Contact Metric Manifolds

Suppose is -conharmonically flat, that is, .

Then from (3.1), we obtain Using (2.9), (2.6), and (2.11) in (3.29), we obtain Taking in (3.30) and using (2.1), we obtain Contracting (3.31) with , we obtain Replacing by in (3.32) and using (2.7) and (2.10), we obtain Tha above equation with (3.32) yields where Hence reduces to an -Einstein manifold.

Thus we have the following.

Theorem 3.4. *A -dimensional -conharmonically flat -contact metric manifold is an -Einstein manifold.*

##### 3.5. -Recurrent -Contact Metric Manifolds

A -dimensional -contact metric manifold is said to be -recurrent if and only if there exists a nonzero 1-form such that Differentiating (3.1) covariantly with respect to , we obtain Suppose is -recurrent. Then from (3.37), we have Contracting (3.38) with , we obtain Since is a nonzero 1-form, we have .

Using (3.1), the above equation yields Using (2.6) and (2.9) in (3.40), we obtain Taking in (3.41), we get Replacing by in (3.42), we obtain Replacing by in (2.10) and comparing the resulting equation with (3.43), we obtain where .

Using (3.44) in (3.42), we get where and .

That is, is an -Einstein manifold.

Thus we have the following.

Theorem 3.5. *A -recurrent -contact metric manifold is an -Einstein manifold.*

#### 4. Conformal Curvature Tensor in -Contact Metric Manifolds

The conformal curvature tensor in is defined by [11]

*Definition 4.1. *A -contact metric manifold is (1)-conformally flat if , (2)conformally Ricci symmetric if , (3)-conformally flat if for all , , , and .

##### 4.1. -Conformally Flat -Contact Metric Manifolds

Suppose that -contact metric manifold is -conformally flat. Then from (4.1), we obtain Using (2.1), (2.6), and (2.11) in (4.2), we obtain Putting in (4.3) and using (2.1) and (2.10), we obtain Contraction of the above with yields From (4.5), we have the following. with and if and only if .

Thus we have the following.

Theorem 4.2. *A -conformally flat -contact metric manifold reduces to -contact metric manifold if and only if it is an -Einstein manifold.*

##### 4.2. Conformally Ricci-Symmetric -Contact Metric Manifolds

If , then we have Taking in (4.7) and using (4.1), (2.6), (2.9) to (2.12), we obtain where Taking in (4.8), we obtain .

If , then , .

Thus for , (4.8) reduces to where that is, reduces to -Einstein.

Thus we have the following.

Theorem 4.3. *A conformally Ricci-symmetric -contact metric manifold is an -Einstein manifold.*

##### 4.3. -Conformally Flat -Contact Metric Manifolds

Suppose is -conformally flat, that is, for all vector fields , , , and . Then from (4.1), we obtain Let be a local orthonormal basis of the tangent space at in .

Taking in (4.12) and summing up from 1 to , we obtain Using (3.22) in (4.13), we obtain Replacing by and by in (4.14) and using (2.1), we have where From the relation (4.15), we conclude that is an -Einstein manifold.

Hence we can state the following.

Theorem 4.4. *A -conformally flat -contact metric manifold is an -Einstein manifold with .*

#### 5. Pseudoprojective Curvature Tensor in -Contact Metric Manifolds

In , the pseudoprojective curvature tensor is given by [11] where and are constants such that , , is the curvature tensor, is the Ricci tensor, is the scalar curvature.

##### 5.1. Pseudoprojectively Symmetric -Contact Metric Manifolds

Suppose holds in . Then we have Taking in (5.2) and using (5.1), (2.5), (2.10), and (2.11), we have Contracting the above with , we obtain Replacing by in (5.4) and using (2.7) and (2.10), we obtain where Substituting for in (5.4), we obtain where From relation (5.7), we conclude that is an -Einstein manifold.

Hence we can state the following.

Theorem 5.1. *A pseudoprojective symmetric -contact metric manifold is an -Einstein manifold.*

##### 5.2. Pseudoprojective Ricci-Symmetric -Contact Metric Manifolds

If , then we have Taking in (5.9) and using (5.1), (2.1), (2.5), (2.10), and (2.11), we obtain where Replacing by in (5.10) and using (2.7) and (2.10), we obtain Now substituting for in (5.10), we obtain where From (5.13), we have that is an -Einstein manifold.

Hence we can state the following.

Theorem 5.2. *A -contact metric manifold is an -Einstein manifold if .*

#### 6. Ricci Semisymmetric -Contact Metric Manifolds

If a -dimensional -contact metric manifold is Ricci semisymmetric, then . That is, Taking in (6.1) and using (2.5), (2.7), and (2.11), we obtain where Replacing by in (6.2) and using (2.7) and (2.10), we obtain Then (6.2) reduces to where From relation (6.5), we conclude that the manifold is an -Einstein manifold.

Hence we can state the following.

Theorem 6.1. *A Ricci semisymmetric -contact metric manifold is an -Einstein manifold.*

#### References

- D. E. Blair, T. Koufogiorgos, and B. J. Papantoniou, “Contact metric manifolds satisfying a nullity condition,”
*Israel Journal of Mathematics*, vol. 91, no. 1–3, pp. 189–214, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - E. Boeckx, “A full classification of contact metric $(k,\mu )$-spaces,”
*Illinois Journal of Mathematics*, vol. 44, no. 1, pp. 212–219, 2000. View at Google Scholar - R. Sharma, “Certain results on
*K*-contact and $(k,\mu )$-contact manifolds,”*Journal of Geometry*, vol. 89, no. 1-2, pp. 138–147, 2008. View at Publisher · View at Google Scholar - B. J. Papantoniou, “Contact manifolds, harmonic curvature tensor and $(k,\mu )$-nullity distribution,”
*Commentationes Mathematicae Universitatis Carolinae*, vol. 34, no. 2, pp. 323–334, 1993. View at Google Scholar - U. C. De and A. K. Gazi, “On $\varphi $-recurrent N($k$)-contact metric manifolds,”
*Mathematical Journal of Okayama University*, vol. 50, pp. 101–112, 2008. View at Google Scholar - H. G. Nagaraja, “On N($k$)-mixed quasi Einstein manifolds,”
*European Journal of Pure and Applied Mathematics*, vol. 3, no. 1, pp. 16–25, 2010. View at Google Scholar - D. E. Blair,
*Contact Manifolds in Riemannian Geometry*, vol. 509 of*Lecture Notes in Mathematics*, Springer, Berlin, Germany, 1976. View at Zentralblatt MATH - D. E. Blair,
*Riemannian Geometry of Contact and Symplectic Manifolds*, vol. 203 of*Progress in Mathematics*, Birkhäuser, Boston, Mass, USA, 2002. View at Zentralblatt MATH - S. Tanno, “Ricci curvatures of contact Riemannian manifolds,”
*The Tohoku Mathematical Journal*, vol. 40, no. 3, pp. 441–448, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - U. C. De and A. A. Shaikh,
*Differential Geometry of Manifolds*, Alpha Science International Ltd, Oxford, UK, 2007. - M. M. Tripathi and P. Gupta, “On $\tau $-curvature tensor in $k$-contact and Sasakian manifolds,”
*International Electronic Journal of Geometry*, vol. 4, no. 1, pp. 32–47, 2011. View at Google Scholar