A characterization of submanifolds by a homogeneity condition

A very short proof of the following smooth homogeneity theorem of D. Repovs, E.V. Shchepin and the author is presented. Let N be a locally compact subset of a smooth manifold M. Assume that for each two points x , y ∈ N there exists a diffeomorphism h : M → M such that h ( x ) = y and h ( N ) = N . Then N is a smooth submanifold of M.