%0 Journal Article
%T On the M-Projective Curvature Tensor of -Contact Metric Manifolds
%A R. N. Singh
%A Shravan K. Pandey
%J ISRN Geometry
%D 2013
%R 10.1155/2013/932564
%X The object of the present paper is to study some curvature conditions on -contact metric manifolds. 1. Introduction The notion of the odd dimensional manifolds with contact and almost contact structures was initiated by Boothby and Wong in 1958 rather from topological point of view. Sasaki and Hatakeyama reinvestigated them using tensor calculus in 1961. Tanno [1] classified the connected almost contact metric manifolds whose automorphism groups possess the maximum dimension. For such a manifold, the sectional curvature of plain sections containing is a constant, say . He showed that they can be divided into three classes: (i) homogeneous normal contact Riemannian manifolds with , (ii) global Riemannian products of line or a circle with a K£¿hler manifold of constant holomorphic sectional curvature if , and (iii) a warped product space if . It is known that the manifolds of class (i) are characterized by admitting a Sasakian structure. Kenmotsu [2] characterized the differential geometric properties of the manifolds of class (iii); so the structure obtained is now known as Kenmotsu structure. In general, these structures are not Sasakian [2]. On the other hand in Pokhariyal and Mishra [3] defined a tensor field on a Riemannian manifold as where and . Such a tensor field is known as m-projective curvature tensor. Later, Ojha [4] defined and studied the properties of m-projective curvature tensor in Sasakian and K hler manifolds. He also showed that it bridges the gap between the conformal curvature tensor, conharmonic curvature tensor, and concircular curvature tensor on one side and H-projective curvature tensor on the other. Recently m-projective curvature tensor has been studied by Chaubey and Ojha [5], Singh et al. [6], Singh [7], and many others. Motivated by the above studies, in the present paper, we study flatness and symmetry property of -contact metric manifolds regarding m-projective curvature tensor. The present paper is organized as follows. In this paper, we study the m-projective curvature tensor of -contact metric manifolds. In Section 2, some preliminary results are recalled. In Section 3, we study m-projectively semisymmetric -contact metric manifolds. Section 4 deals with m-projectively flat -contact metric manifolds. -m-projectively flat -contact metric manifolds are studied in Section 5 and obtained necessary and sufficient condition for an -contact metric manifold to be -m-projectively flat. In Section 6, m-projectively recurrent -contact metric manifolds are studied. Section 7 is devoted to the study of -contact metric manifolds
%U http://www.hindawi.com/journals/isrn.geometry/2013/932564/