Suppose we have a real /^-dimensional <^2 manifold M embedded in C n . If M has a nondegenerate complex tangent bundle of positive rank at some point pe M 9 then the vanishing or nonvanishing of the Levi form on M near p determines whether or not M is locally holomorphic at p. We show that if M is locally holomorphic at p, then the Levi form vanishes near p, the converse being a known result. In addition we prove a C -R extendibility theorem for a certain case when M is ^~ and has a nonzero Levi form at peM.