Some Results on Generalized Quasi-Einstein Manifolds

This paper deals with generalized quasi-Einstein manifold satisfying certain conditions on conharmonic curvature tensor. Here we study some geometric properties of generalized quasi-Einstein manifold and obtain results which reveal the nature of its associated 1-forms. 1. Introduction It is well known that a Riemannian or a semi-Riemannian manifold ？？ is an Einstein manifold if its Ricci tensor of type is of the form , being the (constant) scalar curvature of the manifold. Let . Then the manifold is said to be a quasi-Einstein manifold [1–7] if, on , we have where is -form on and , are some smooth functions on . It is clear that the function and the -form are nonzero at every point on . The scalars , are the associated scalars of the manifold. Also the 1-form is called associated 1-form of the manifold defined by for any vector field , being a unit vector field called generator of the manifold. Such an -dimensional quasi-Einstein manifold is denoted by . The quasi-Einstein manifolds have also been studied in [8–11]. Generalizing the notion of quasi-Einstein manifold, in [12], De and Ghosh introduced the notion of generalized quasi-Einstein manifolds and studied its geometrical properties with the existence of such notion by several nontrivial examples. A Riemannian manifold ？？ is said to be generalized quasi-Einstein manifold if its Ricci tensor of type is not identically zero and satisfies the condition: where , , and？？ are scalars of which , and , are nowhere vanishing 1-forms such that , for any vector field . The unit vectors and corresponding to 1-forms and are orthogonal to each other. Also and are the generators of the manifold. Such an -dimensional manifold is denoted by . The generalized quasi-Einstein manifolds have also been studied in [13–16]. In 2008, De and Gazi [17] introduced the notion of nearly quasi-Einstein manifolds. A nonflat Riemannian manifold ？？ is called a nearly quasi-Einstein manifold if its Ricci tensor of type is not identically zero and satisfies the condition: where and are nonzero scalars and is a nonzero symmetric tensor of type . An -dimensional nearly quasi-Einstein manifold was denoted by . The nearly quasi-Einstein manifolds have also been studied by Prakasha and Bagewadi [18]. The present paper is organized as follows. Section 2 deals with the preliminaries. Section 3 is concerned with conharmonic curvature tensor on . In this section, it is proved that a conharmonically flat is one of the manifold of generalized quasiconstant curvature. Also, it is proved that if in a ？？ the associated scalars are constants and