A new class of contact Riemannian manifolds

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) (<(del)over cap>((gamma) over d ot) R)(., (gamma) over dot) (gamma) over dot = 0 for any unit <(del)over ca p>-geodesic gamma(<(del)over cap>((gamma) over dot) (gamma) over dot = 0), where <(del)over cap> 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).