Algebraic prolongation and rigidity of Carnot groups

We discuss the known results on rigidity of Carnot groups using Tanaka’s prolongation theory. We also apply Tanaka’s theory to study rigidity of an extended class of H-type groups which we call J-type groups. In particular we obtain a rigidity criterion giving rise to a rigid class of J-type groups which includes the H-type groups, and thus extends the results of H.M. Reimann. We also construct a noncomplex J-type group which is nonrigid and does not satisfy the rank 1 condition over the reals.