Studies on a Type of Para-Kenmotsu Manifold

In this chapter, we consider a class of almost para-contact metric manifold namely para-Kenmotsu (briefly P-Kenmotsu) manifold Mn admitting the condition R(X, Y).C = 0 where C is the conformal curvature tensor of the manifold and R is the Riemannian curvature tensor. R(X, Y) is considered as a derivation of the tensor algebra at each point of the manifold for tangent vectors X and Y. We study and have shown that a P-Kenmotsu manifold (Mn, g) (n > 3) admitting the condition R(X, Y).C = 0 is conformally flat and hence is an SP-Kenmotsu manifold, where ‘g’ is the Riemannian metric. For a conformally symmetric Riemannian manifold, we have and hence for such a manifold R(X, Y).C = 0 holds. Thus we have the following corollary. It says that a conformally symmetric P-Kenmotsu manifold (Mn, g) (n > 3) is an SP-Kenmotsu manifold. The chapter ends with a concluding remark that to identify and strengthen the physical significance of the structures and connections discussed in this chapter.