  Let M,M' be smooth real hypersurfaces in N-dimensional space and assume that
M is k-nondegenerate at a point p in M. We prove that holomorphic mappings that
extend smoothly to M, sending a neighborhood of p in M diffeomorphically into
M' are completely determined by their 2k-jet at p. As an application of this
result, we also give sufficient conditions on a smooth real hypersurface which
guarantee that the space of infinitesimal CR automorphisms is finite
dimensional.
