On the Conharmonic Curvature Tensor of Generalized Sasakian-Space-Forms

The object of the present paper is to characterize generalized Sasakian-space-forms satisfying certain curvature conditions on conharmonic curvature tensor. In this paper we study conharmonically semisymmetric, conharmonically flat, -conharmonically flat, and conharmonically recurrent generalized Sasakian-space-forms. Also generalized Sasakian-space-forms satisfying and have been studied. 1. Introduction Conformal transformations of a Riemannian structures are an important object of study in differential geometry. Of considerable interest in a special type of conformal transformations, conharmonic transformations, which are conformal transformations are preserving the harmonicity property of smooth functions. This type of transformation was introduced by Ishii [1] in 1957 and is now studied from various points of view. It is well known that such transformations have a tensor invariant, the so-called conharmonic curvature tensor. It is easy to verify that this tensor is an algebraic curvature tensor; that is, it possesses the classical symmetry properties of the Riemannian curvature tensor. Let and be two Riemannian manifolds with and being their respective metric tensors related through where is a real function. Then and are called conformally related manifolds, and the correspondence between and is known as conformal transformation [2]. It is known that a harmonic function is defined as a function whose Laplacian vanishes. A harmonic function is not invariant, in general. The conditions under which a harmonic function remains invariant have been studied by Ishii [1] who introduced the conharmonic transformation as a subgroup of the conformal transformation (1.1) satisfying the condition where comma denotes the covariant differentiation with respect to metric . A rank-four tensor that remains invariant under conharmonic transformation for -dimensional Riemannian manifold is given by where and denote the Riemannian curvature tensor of type defined by and the Ricci tensor of type , respectively. The curvature tensor defined by (1.3) is known as conharmonic curvature tensor. A manifold whose conharmonic curvature tensor vanishes at every point of the manifold is called conharmonically flat manifold. Thus this tensor represents the deviation of the manifold from canharmonic flatness. Conharmonic curvature tensor has been studied by Abdussattar [3], Siddiqui and Ahsan [2], ？zgür [4], and many others. Let be an almost contact metric manifold equipped with an almost contact metric structure . At each point , decompose the tangent space into the direct sum ,