Chinese Journal of Mathematics

Volume 2014, Article ID 736846, 5 pages

http://dx.doi.org/10.1155/2014/736846

## On Generalized -Recurrent -Trans-Sasakian Manifolds

^{1}Department of Mathematics, Kuvempu University, Jnana Sahyadri, Shimoga 577451, India^{2}Department of Mathematics, B.V.B. College of Engineering & Technology, Hubli 580 032, India

Received 14 September 2013; Accepted 17 December 2013; Published 30 January 2014

Academic Editors: P. Li and C. Yin

Copyright © 2014 C. S. Bagewadi and Dakshayani A. Patil. 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 generalized -recurrent -trans-Sasakian manifolds. A relation between the associated 1-forms and and relation between characteristic vector field *ξ* and the vector fields for a generalized -recurrent.

#### 1. Introduction

As a generalization of both Sasakian and Kenmotsu manifolds, Oubiña [1] introduced the notion of trans-Sasakian manifolds, which are closely related to the locally conformal Kahler manifolds. Trans-Sasakian manifolds of types ,, and are, respectively, called the cosymplectic, -Sasakian, and -Kenmotsu manifolds, with , being scalar functions. In particular, if,;,; then a trans-Sasakian manifold will be Kenmotsu and Sasakian manifolds respectively. The local structure of trans-Sasakian manifolds of dimension has been completely characterized by Marrero [2]. He proved that a trans-Sasakian manifold of dimension is either cosymplectic or -Kenmotsu or -Sasakian manifold.

The investigation of manifolds with indefinite metrics was taken by several authors. In 1993, Bejancu and Duggal [3] introduced the concept of -Sasakian manifolds and Xufeng and Xiaoli [4] established that these manifolds are real hypersurfaces of indefinite Kahlerian manifolds. Kumar et al. [5] studied the curvature conditions of these manifolds. Tripathi et al. [6] introduced and studied -almost paracontact manifolds.

The paper is organised as follows: Section 2 contains preliminaries of -trans-Sasakian manifolds. In Sections 3 and 4 we obtain results for generalized -recurrent and generalized concircular -recurrent -Trans-Sasakian Manifolds.

#### 2. Preliminaries

Let be the connected almost contact metric manifold with an almost contact metric structure ; that is, is a tensor field, is a vector field, is a -form, and is compatible Riemannian metric, such that Almost contact metric manifold is called -almost contact metric manifold if for all vector fields,on, where .

An -almost contact metric manifold is called a -trans-Sasakian manifold if holds for some smooth functions and on and , . For , , -trans-Sasakian manifold reduces to -Sasakian and for , it reduces to -Kenmotsu manifold. -Trans-Sasakian manifolds have been studied by authors [7] and they obtained the following results:

#### 3. Generalized -Recurrent -Trans-Sasakian Manifolds

*Definition 1. *-trans-Sasakian manifold is called generalized -recurrent if its curvature tensor satisfies the condition
where and are two -forms, is nonzero, and these are defined by
and , are vector fields associated with -forms and , respectively.

Let us consider a generalized -recurrent -trans-Sasakian manifold. Then by virtue of (1) and (13) we have from which it follows that Let , , be an orthonormal basis of the tangent space at any point of the manifold. Then putting in (16) and taking summation over, , we get Put in (17) and by virtue of (8) and the equation , we have Now we have Using (5) and (6) in the above relation it follows that From (18) and (20) we obtain Replacing in (21) and then using (2) we get

Theorem 2. *In a generalized -recurrent -trans-Sasakian manifold, the -forms and are related as in (22).*

If , then, from (22), we obtain

Corollary 3. *In a generalized -recurrent --Sasakian manifold, the -forms and are related as in (23).*(i)*If , then --Sasakian manifold reduces as -Sasakian manifold.*(ii)*If and , then --Sasakian manifold reduces as Sasakian manifold.*

*If , then, from (22), we obtain
*

*Corollary 4. In a generalized -recurrent --Kenmotsu manifold, the -forms and are related as in (24).*

*If , then, from (22), we obtain
From definition (13) this leads to the following corollary.*

*Corollary 5. In cosymplectic manifold, generalized -recurrent reduces to -recurrent manifold.*

*Now from (15) we have
From (26) and second Bianchi identity we get
By virtue of (12) we obtain from (27) that
Putting in (28) and taking summation over , we get
If , then, from (29), we obtain
for all vector fields ,. Replacing by in (30), we get
for all vector field .*

*From (30) and (31), we can state the following.*

*Theorem 6. A generalized -recurrent --Sasakian manifold, the characteristic vector field , and the vector fields , associated with the -forms ,, respectively, are in the same direction and the -forms , are given by (31).*

*4. Generalized Concircular -Recurrent -Trans-Sasakian Manifolds*

*In this section we study three-dimensional generalized concircular -recurrent -trans-Sasakian manifold.*

*Definition 7. *-Trans-Sasakian manifold is called generalized concircular -recurrent if its concircular curvature tensor

*
satisfies the condition
where and are defined as in (14) and is the scalar curvature of the manifold .*

*Let us consider a generalized concircular -recurrent -trans-Sasakian manifold. Then by virtue of (1) we have
from which it follows that
Let ,, be a local orthonormal basis of the tangent space at any point of the manifold. Then putting in (35) and taking summation over, , we get
Putting in (36) and using (11) we have
This leads to the following.*

*Theorem 8. In a generalized concircular -recurrent -trans-Sasakian manifold , the -forms and are related as in (37).*

*Conflict of Interests*

*The authors declare that there is no conflict of interests regarding the publication of this paper.*

*References*

- J. A. Oubiña, “New classes of almost contact metric structures,”
*Publicationes Mathematicae Debrecen*, vol. 32, no. 3-4, pp. 187–193, 1985. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. C. Marrero, “The local structure of trans-Sasakian manifolds,”
*Annali di Matematica Pura ed Applicata*, vol. 162, pp. 77–86, 1992. View at Publisher · View at Google Scholar - A. Bejancu and K. L. Duggal, “Real hypersurfaces of indefinite Kaehler manifolds,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 16, no. 3, pp. 545–556, 1993. View at Publisher · View at Google Scholar - X. Xufeng and C. Xiaoli, “Two theorems on $(\u03f5)$-Sasakian manifolds,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 21, no. 2, pp. 249–254, 1998. View at Publisher · View at Google Scholar · View at MathSciNet - R. Kumar, R. Rani, and R. K. Nagaich, “On sectional curvatures of $(\u03f5)$-Sasakian manifolds,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 2007, Article ID 93562, 8 pages, 2007. View at Publisher · View at Google Scholar - M. M. Tripathi, E. Kılıç, S. Y. Perktaş, and S. Keleş, “Indefinite almost paracontact metric manifolds,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 2010, Article ID 846195, 19 pages, 2010. View at Publisher · View at Google Scholar - H. G. Nagaraja, R. C. Premalatha, and G. Somashekara, “On an $(\u03f5,\delta )$-trans-Sasakian structure,”
*Proceedings of the Estonian Academy of Sciences*, vol. 61, no. 1, pp. 20–28, 2012. View at Publisher · View at Google Scholar · View at MathSciNet

*
*