A characterization of submanifolds by a homogeneity condition
  A very short proof of the following smooth homogeneity theorem of D. Repovs, E. V. Scepin 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 in N there exist their neighborhoods Ux and Uy in M and a diffeomorphism h : Ux \to Uy such that h(x)=y and h (Ux \cap N) = Uy \cap N. Then N is a smooth submanifold of M. 
A. Skopenkov