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ISRN Geometry
Volume 2013 (2013), Article ID 412593, 6 pages
http://dx.doi.org/10.1155/2013/412593
Research Article

A Study on Ricci Solitons in Kenmotsu Manifolds

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

Received 27 December 2012; Accepted 16 January 2013

Academic Editors: R. Farnsteiner and V. S. Matveev

Copyright © 2013 C. S. Bagewadi 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 study and obtain results on Ricci solitons in Kenmotsu manifolds satisfying , , , and , where and are C-Bochner and pseudo-projective curvature tensor.

1. Introduction

A Ricci soliton is a natural generalization of an Einstein metric and is defined on a Riemannian manifold . A Ricci soliton is a triple with a Riemannian metric, a vector field, and a real scalar such that where is a Ricci tensor of and denotes the Lie derivative operator along the vector field . The Ricci soliton is said to be shrinking, steady, and expanding accordingly as is negative, zero, and positive, respectively [1]. In this paper, we prove conditions for Ricci solitons in Kenmotsu manifolds to be shrinking, steady, and expanding.

In 1972, Kenmotsu [2] studied a class of contact Riemannian manifolds satisfying some special conditions and this manifold is known as Kenmotsu manifolds. Kenmotsu proved that a locally Kenmotsu manifold is a warped product of an interval and a Kaehler manifold with warping function , where is a nonzero constant. Kenmotsu proved that if in a Kenmotsu manifold the condition holds, then the manifold is of negative curvature , where is the curvature tensor of type and denotes the derivation of the tensor algebra at each point of the tangent space.

The authors in [37] have studied Ricci solitons in contact and Lorentzian manifolds. The authors in [8] have obtained some results on Ricci solitons satisfying , , and and now we extend the work to , , , and .

2. Preliminaries

An -dimensional differential manifold is said to be an almost contact metric manifold [9] if it admits an almost contact metric structure consisting of a tensor field of type , a vector field , a -form , and a Riemannian metric compatible with satisfying for all vector fields , on .

An almost contact metric manifold is said to be Kenmotsu manifold [2] if From (3), we have where denotes the Riemannian connection of .

In an -dimensional Kenmotsu manifold, we have where is the Riemannian curvature tensor.

Let be a Ricci soliton in an -dimensional Kenmotsu manifold . From (4) we have From (1) and (9) we get

The above equation yields that where is the Ricci tensor, is the Ricci operator, and is the scalar curvature on .

2.1. Example for 3-Dimensional Kenmotsu Manifolds

We consider -dimensional manifold , where are the standard coordinates in . Let be linearly independent given by

Let be the Riemannian metric defined by , , where is given by

The structure is given by

The linearity property of and yields that , , , for any vector fields on . By definition of Lie bracket, we have

Let be the Levi-Civita connection; with respect to above metric is given by Koszula formula and by virtue of it we have

Clearly (19) shows that satisfies (2), (3), and (4). Thus is a Kenmotsu manifold.

It is known that With the help of (19) and (20), it can be easily verified that

From the above expression of the curvature tensor we obtain

Similarly we have

Now by , in (1), where and by virtue of above equations we get the value of which is strictly greater than . Thus this is an example of expanding Ricci solitons in Kenmotsu manifolds.

3. Ricci Soliton in a Kenmotsu Manifold Satisfying

Bochner introduced a Kähler analogue of the Weyl conformal curvature tensor by purely formal considerations, which is now well known as the Bochner curvature tensor [10]. A geometric meaning of the Bochner curvature tensor is given by Blair in [11] by using the Boothby-Wang's fibration. In 1969, Matsumoto and Chūman [12] constructed the notion of C-Bochner curvature tensor in a Sasakian manifold and studied its several properties.

The C-Bochner curvature tensor [13] in is defined by where .

Taking in (24) and using (6), (10), (11), we get

Similarly using (5), (10), (11), (12) in (24), we get

We assume that the condition , then we have Using (7) in (27), we get

By taking an inner product with , we have

By using (25), (26) in (29), we have

In view of (24) in (30), then we have

Taking in (31) and summing over . By virtue of (10), (11), (12), and on simplification, we get

Putting in (32) and by virtue of (10) and (13), we have

Therefore, positive that is, the Ricci soliton in Kenmotsu manifold is expanding.

Hence we state the following theorem:

Theorem 1. A Ricci soliton in a Kenmotsu manifold satisfying is expanding.

4. Ricci Soliton in a Kenmotsu Manifolds Satisfying

The condition implies that

By using (10) in (34), we have the above equation implies that

By using (24) and (26) in (36), we have

Put in (37) then the equation is identically satisfied and we do not get the value for . So, we proceed as follows: Taking in (37) and summing over and by virtue of (13) and conditions, we obtain

Therefore, is positive that is Ricci soliton in Kenmotsu manifolds satisfying is expanding.

Hence we can state the following theorem.

Theorem 2. A Ricci soliton in a Kenmotsu manifold satisfying is expanding.

5. Ricci Soliton in a Kenmotsu Manifold Satisfying

Using the following equations: where the endomorphism is defined by

we have

By using the condition , and by virtue of (10), (12), we have

By taking an inner product with and by virtue of (5), (6), (7), and (8), we have

Taking and summing over , we obtain

Taking in (44) and by virtue of (12), (13), we obtain

This implies either

Therefore for any or the Ricci soliton in Kenmotsu manifolds satisfying is either steady or shrinking.

Hence we can state the following theorem.

Theorem 3. A Ricci soliton in a Kenmotsu manifold satisfying is either steady or shrinking.

6. Ricci Soliton in a Kenmotsu Manifolds Satisfying

The Pseudo-projective curvature tensor is defined by where are constants. Taking in (47) and using (6), (10), (11), we get

Similarly using (5), (10), (11), (12) in (47), we get

We assume that the condition , then we have

Using (7) in (50), we find

By taking an inner product with then we get

By using (48), (49) in (52), we have

In view of (47) in (53), we have

Taking in (54) and summing over , and on simplification, we get

Putting in (55) and by virtue of (12), (13), we get the following equation:

Since implies that , that is, the Ricci soliton in Kenmotsu manifold satisfying is expanding,

hence we state the following theorem.

Theorem 4. A Ricci soliton in a Kenmotsu manifold satisfying is expanding.

7. Ricci Soliton in a Kenmotsu Manifolds Satisfying

The condition implies that

By using (10) in (57), we have that is,

By using (47) and (48) in (59), we have

Put in (60); then the equation is identically satisfied and we do not get the value for . So, we proceed as follows: taking , summing over , and by virtue of (13) and conditions we obtain

Therefore, if in (61) then ; that is, Ricci soliton in Kenmotsu manifolds satisfying is steady. If then either for or for , that is, the Ricci soliton in Kenmotsu manifold satisfying is expanding or shrinking.

Hence we can state the following theorem.

Theorem 5. A Ricci soliton in a Kenmotsu manifolds satisfying is steady for , expanding for and shrinking for .

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