On Generalized Sasakian-Space-Forms

The purpose of the present paper is to characterize pseudoprojectively flat and pseudoprojective semisymmetric generalized Sasakian-space-forms. 1. Introduction Alegre et al. [1] introduced and studied the generalized Sasakian-space-forms. The authors Alegre and Carriazo [2], Somashekhara and Nagaraja [3, 4], and De and Sarkar [5, 6] studied the generalized Sasakian-space-forms. An almost contact metric manifold is said to be a generalized Sasakian-space-form if there exist differentiable functions such that curvature tensor of is given by for any vector fields , , on , where In this paper, we study the curvature properties like flatness, symmetry, and semisymmetry properties in a generalized Sasakian-space-form by considering a pseudoprojective curvature tensor. The paper is organized as follows. Section 2 of this paper contains some preliminary results on the generalized Sasakian-space-forms. In Section 3, we study the pseudoprojectively flat generalized Sasakian-space-form and obtain necessary and sufficient conditions for a generalized Sasakian-space-form to be pseudoprojectively flat. In the next section, we deal with pseudoprojectively semisymmetric generalized Sasakian-space-forms, and it is proved that a generalized Sasakian-space-form is pseudoprojectively semisymmetric if and only if the space form is pseudoprojectively flat and . The last section is devoted to the study of -flat and -semi symmetric generalized Sasakian-space-forms. In this section, we prove that the associated functions are linearly dependent. In a -dimensional almost contact metric manifold, the pseudoprojective curvature tensor [7] is defined by where and are constants and , , and are the Riemannian curvature tensor of type , the Ricci tensor, and the scalar curvature of the manifold, respectively. If ,？？ , then (1.3) takes the form where is the projective curvature tensor. A manifold shall be called pseudoprojectively flat if the pseudoprojective curvature tensor . It is known that the pseudoprojectively flat manifold is either projectively flat (if ) or Einstein (if and ). 2. Preliminaries A -dimensional -differentiable manifold is said to admit an almost contact metric structure if it satisfies the following relations: where is a tensor field of type , is a vector field, is a 1-form, and is a Riemannian metric on . A manifold equipped with an almost contact metric structure is called an almost contact metric manifold. An almost contact metric manifold is called a contact metric manifold if it satisfies for all vector fields and . In a generalized Sasakian-space-form, the