On 3-Dimensional Contact Metric Generalized -Space Forms

The present paper deals with a study of 3-dimensional contact metric generalized -space forms. We obtained necessary and sufficient condition for a 3-dimensional contact metric generalized -space form with to be of constant curvature. We also obtained some conditions of such space forms to be pseudosymmetric and -projectively flat, respectively. 1. Introduction In 1995, Blair et al. [1] introduced the notion of contact metric manifolds with characteristic vector field belonging to the -nullity distribution and such types of manifolds are called -contact metric manifolds. They obtained several results and examples of such a manifold. A full classification of this manifold has been given by Boeckx [2]. A contact metric manifold is said to be a generalized -space if its curvature tensor satisfies the condition for some smooth functions and on independent choice of vector fields and . If and are constant, the manifold is called a -space. If a -space has a constant -sectional curvature and a dimension greater than 3, the curvature tensor of this -space form is given by [3] where , , , , , and are the tensors defined by for all vector fields , , on , where and is the usual Lie derivative. The notion of generalized Sasakian-space-forms was introduced and studied by Alegre et al. [4] with several examples. A generalized Sasakian-space-form is an almost contact metric manifold whose curvature tensor is given by where , , and are the tensors defined above and , , are differentiable functions on . In such case we will write the manifold as . Generalized Sasakian-space-forms have been studied by several authors, namely, [5–11]. By motivating the works on generalized Sasakian-space-forms and -space forms, Carriazo et al. [12] introduced the concept of generalized -space forms. A generalized -space form is an almost contact metric manifold whose curvature tensor is given by where , , , , , are the tensors defined above and , , , , , are differentiable functions on . The object of the paper is to study 3-dimensional contact metric generalized -space forms. The paper is organized as follows. Section 2 deals with some preliminaries on contact metric manifolds and contact metric generalized -space forms. Section 3 is concerned with 3-dimensional contact metric generalized -space forms. Here it is proved that a 3-dimensional contact metric generalized -space form with is of constant curvature if and only if . In Section 4, it is proved that a 3-dimensional contact metric generalized -space form is Ricci-semisymmetric; then either or . In Sections 5 and 6, we obtained that