Abstract: 1. Introduction.This paper is a continuation of the author's paper [1] in which it was shown that there are no flat contact metric structures on a contact manifold of dimension ^ 5.Although this result is a non-existence theorem, the argument yields some positive results on certain contact metric manifolds.Here we prove two such results.THEOREM A. A contact metric manifold M 2n+1 is a K-contact manifold if and only if the Ricci curvature in the direction of the characteristic vector field ξ is equal to 2n.THEOREM B. Let M 2n+1 be a contact metric manifold and suppose that R(X, Y)ξ = 0 for all vector fields X and Y. Then M 2n+1 is locally the product of a fiat (n + ΐ)-dimensional manifold and an n-dimensional manifold of positive constant curvature 4.