A new class of contact riemannian manifolds

Abstract

N. Tanaka ([10]) defined the canonical affine connection on a nondegenerate integrable CR manifold. In the present paper, we introduce a new class of contact Riemannian manifolds satisfying (C) ( $$(C)(\hat \nabla \dot \gamma R)( \cdot ,\dot \gamma )\dot \gamma = 0$$ for any unit $$\hat \nabla $$ -geodesic ( $$\gamma (\hat \nabla _{\dot \gamma } \dot \gamma = 0)$$ , where $$\hat \nabla $$ is the generalized Tanaka connection. In particular, when the associated CR structure of a given contact Riemannian manifold is integrable we have a structure theorem and find examples which are neither Sasakian nor locally symmetric but satisfy the condition (C).