ISRN Geometry

Volume 2013 (2013), Article ID 392608, 7 pages

http://dx.doi.org/10.1155/2013/392608

Research Article

-Homothetic Deformation of -Contact Manifolds

H. G. Nagaraja and C. R. Premalatha

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

Received 12 October 2013; Accepted 11 November 2013

Academic Editors: A. M. Cegarra and J. L. Cieśliński

Copyright © 2013 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 -homothetic deformations of -contact manifolds. We prove that -homothetically deformed -contact manifold is a generalized Sasakian space form if it is conharmonically flat. Further, we find expressions for scalar curvature of -homothetically deformed -contact manifolds.

1. Introduction

In 1968 Tanno [1] introduced the notion of -homothetic deformations. Carriazo and Martín-Molina [2] studied -homothetic deformation of generalized space forms and gave several examples for manifolds of dimension 3. De and Ghosh [3] studied -homothetic deformation of almost normal contact metric manifolds and prove that is invariant under such transformation. Bagewadi and Venkatesha [4] studied concircularly semisymmetric trans-Sasakian manifolds and De et al. [5] studied conharmonically semisymmetric, conharmonically flat, -conharmonically flat, and conharmonically recurrent generalized Sasakian space forms. Several authors [6–11] studied -contact manifolds and proved conditions for these manifolds to be of -conformally flat, -conformally flat, quasi-conharmonically flat, and -conharmonically flat. Motivated by the above studies, in this paper we study -homothetic deformations of -contact manifolds by considering conharmonic and projective curvature tensor. The paper is organized as follows. After Preliminaries, we give a brief account of information of -homothetic deformation of -contact manifolds in Section 3. In Section 4, we study conharmonically flat, semisymmetric, -conharmonically flat, quasi-conharmonically flat, and -conharmonically flat -contact manifolds with respect to -homothetic deformation. In the last section, we consider Weyl projective curvature in -contact manifolds with respect to -homothetic deformation.

2. Preliminaries

Let be a -dimensional almost contact metric manifold [12], consisting of a tensor field , a vector field , a 1-form , and Riemannian metric . Then

for all , . If is a Killing vector field, then is called a -contact Riemannian manifold [13]. A -contact Riemannian manifold is called Sasakian [12], if the relation

holds, where denotes the operator of covariant differentiation with respect to .

If is a -contact Riemannian manifold, then besides (1), (2), (3), and (4) the following relations hold [14]:

for any vector fields and , where and denote, respectively, the curvature tensor of type and the Ricci tensor of type .

Definition 1. A contact metric manifold is said to be -Einstein if , where and are smooth functions on .

3. -Homothetic Deformation of -Contact Manifolds

Let be a -dimensional almost contact metric manifold. A -homothetic deformation is defined by

with being a positive constant [1].

It is clear that the is also an almost contact metric manifold.

If is a -contact manifold with Riemannian connection , the connection of the -deformed -contact manifold can be calculated from and . Using Koszul's formula and (5), (6), and (11), of is given by

Using (12), we obtain

The curvature tensor of is given by

Using (9), (10), and (14), we have

From (14), we get

where and are the Ricci tensors of and , respectively.

It follows from (16) that

Again contracting (16) over , , we get

where and are the scalar curvatures of and , respectively.

4. Conharmonic Curvature Tensor in -Homothetically Deformed -Contact Manifolds

The conharmonic tensor of a -homothetically deformed -contact manifold is defined by [15]

for , , , where , , and are the Riemannian curvature tensor, Ricci tensor, and Ricci operator of .

Definition 2. An almost contact metric manifold is said to be

(1)conharmonically flat if

(2)conharmonically semisymmetric if

(3)-conharmonically flat if

(4)quasi-conharmonically flat if

(5)-conharmonically flat if

for all vector fields , , and .

Assume that is conharmonically flat -contact manifold with respect to -homothetic deformation. So, we have .

Then from (20), we have

Setting , contracting (26) with , and using (7), (9), (14), and (16), we obtain

Taking in (27) and using (1), (7), and (16), it follows that

Thus, is -Einstein.

Using (28) in (26), we obtain

From (29), we get

Hence, it reduces to a generalized Sasakian space form with , , and . Thus, (30) leads to the following.

Theorem 3. A conharmonically flat -contact manifold admitting -homothetic deformation reduces to a generalized Sasakian space form with associated functions , , and .

Let us now consider a conharmonically semisymmetric -contact manifold admitting -homothetic deformation. Then the condition

holds on for all vector fields , .

From (8), (14), (17), and (20), we obtain

Setting , in (32), we get

Again taking in (32) and using (17), we obtain

Now, (21) yields

Therefore,

From this it follows that

where

Taking in (37) and making use of (32) and (33), we obtain

If is a local orthonormal basis of vector fields in , then, from (39), we get

From (20), it follows that

Using (41) in (40), we obtain

where

In view of (34), (42) yields

where

Thus, is -Einstein.

If is a local orthonormal basis of vector fields in , then, from (44), we get

where

So, we can state the following.

Theorem 4. In a -dimensional conharmonically semisymmetric -contact manifold admitting -homothetic deformation, scalar curvature is given by (46).

Analogous to the definition of -conharmonically flat -contact manifolds [8], we define -conharmonically flat -contact manifolds with respect to -homothetic deformation. Let us assume that is a -conharmonically flat -contact manifold with respect to -homothetic deformation. It can be easily seen that

where , , , .

Using (20), (48) yields

for all , , , .

If is a local orthonormal basis of vector fields in , then is also a local orthonormal basis. So, using (1), (6), (14), (16), and (18), it can be easily verified that

For a local orthonormal basis of vector fields in , putting in (49) and summing up with respect to , we have

for all , . The previous equation, in view of (50), becomes

for all , .

Using (2) and (18), (52) reduces to

Setting in (53), summing up with respect to , and using (19), we obtain

Replacing by and by in (53) and using (54), we obtain for all .

Now using the previous expression in (49), we obtain

for all , , , .

The converse is obvious. Thus we have the following.

Theorem 5. A -dimensional -contact manifold is -conharmonically flat with respect to -homothetic deformation if and only if satisfies (55).

From (20), we obtain

for all , , , .

Suppose that is quasi-conharmonically flat -contact manifold with respect to -homothetic deformation; that is,

Then (56) reduces to

For a local orthonormal basis of vector fields in , putting and in (58) and summing up with respect to , we obtain

Using (2), (10), (14), and (17) in (59), we obtain

Taking and using (1) and (17), we obtain

and using (61) in (19), we obtain

Hence, we can state the following.

Theorem 6. Let be a -contact manifold. Suppose that is obtained from by -homothetic deformation. If is quasi-conharmonically flat, then the scalar curvatures and of and are, respectively, given by (61) and (62).

Suppose that is -conharmonically flat. Then from (20), we have

Contracting the above equation with respect to , we obtain

for all , , , .

For a local orthonormal basis of vector fields in , using (64), we obtain

Therefore,

Using (17) in (66), we obtain

Taking in (64) and using (10), (14), and (17), we obtain

From this we can conclude that is -Einstein. Thus, we have the following.

Theorem 7. A -Homothetically deformed -conharmonically flat -contact manifold is -Einstein and its scalar curvature vanishes.

5. -Homothetic Deformation of -Weyl Projectively Flat -Contact Manifolds

Suppose that, in a -dimensional -contact manifold with -homothetic deformation, the Ricci tensor vanishes; that is,

Then from (16), we have

The Weyl projective curvature tensor of is given by [16]

If , then (71) reduces to

Using (70) in (72), we obtain

The Weyl projective curvature tensor of with respect to -homothetic deformation is given by

Now using (14) and (69) in (74), we get

From (73), the (75) reduces to

Thus, we can state the following.

Theorem 8. Let be obtained from a -contact manifold by -homothetic deformation. If the Ricci tensor of vanishes, then it is -Weyl projectively flat.

References

S. Tanno, “The topology of contact Riemannian manifolds,” Illinois Journal of Mathematics, vol. 12, pp. 700–717, 1968.

A. Carriazo and V. Martín-Molina, “Generalized -space forms and -homothetic deformations,” Balkan Journal of Geometry and its Applications, vol. 16, no. 1, pp. 37–47, 2011.

U. C. De and S. Ghosh, “D-homothetic deformation of normal almost contact metric manifolds,” Ukrainian Mathematical Journal, vol. 64, no. 10, pp. 1514–1530, 2013.

C. S. Bagewadi and Venkatesha, “Some curvature tensors on a trans-Sasakian manifold,” Turkish Journal of Mathematics, vol. 31, no. 2, pp. 111–121, 2007.

U. C. De, R. N. Singh, and S. K. Pandey, “On the conharmonic curvature tensor of generalized Sasakian-space-forms,” ISRN Geometry, vol. 2012, Article ID 876276, 14 pages, 2012.

Z. Guo, “Conformally symmetric -contact manifolds,” Chinese Quarterly Journal of Mathematics, vol. 7, no. 1, pp. 5–10, 1992.

G. Zhen, J. L. Cabrerizo, L. M. Fernández, and M. Fernández, “On -conformally flat contact metric manifolds,” Indian Journal of Pure and Applied Mathematics, vol. 28, no. 6, pp. 725–734, 1997.

J. L. Cabrerizo, L. M. Fernández, M. Fernández, and Z. Guo, “The structure of a class of -contact manifolds,” Acta Mathematica Hungarica, vol. 82, no. 4, pp. 331–340, 1999.

M. K. Dwivedi, L. M. Fernández, and M. M. Tripathi, “The stucture of some classes of contact metric manifolds,” Georgian Mathematical Journal, vol. 16, no. 2, pp. 295–304, 2009.

M. M. Tripathi and M. K. Dwivedi, “The structure of some classes of -contact manifolds,” Proceedings of the Indian Academy of Sciences, vol. 118, no. 3, pp. 371–379, 2008.

M. K. Dwivedi and J.-S. Kim, “On conharmonic curvature tensor in -contact and Sasakian manifolds,” Bulletin of the Malaysian Mathematical Sciences Society, vol. 34, no. 1, pp. 171–180, 2011.

D. E. Blair, Contact Manifolds in Riemannian Geometry, vol. 509 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1976.

M. C. Chaki and M. Tarafdar, “On a type of Sasakian manifold,” Soochow Journal of Mathematics, vol. 16, no. 1, pp. 23–28, 1990.

J. A. Oubiña, “New classes of almost contact metric structures,” Publicationes Mathematicae Debrecen, vol. 32, no. 3-4, pp. 187–193, 1985.

Y. Ishii, “On conharmonic transformations,” The Tensor Society, vol. 7, pp. 73–80, 1957.

R. S. Mishra, Structures on a Differentiable Manifold and Their Applications, Chandrama Prakashan, 50-A, Bairampur House, Allahabad, India, 1984.