-Homothetic Deformation of -Contact Manifolds

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