Conformal Rigidity

Abstract

In one of his less known papers, Cartan [3] studies the conformal deformations of hypersurfaces of an Euclidean space $R^{n+1},n>4.$As a consequence of his methods, he obtains a (local) sufficient condition for conformal rigidity ([3], pg. 101; see also Corollary 1.3 below). In this paper we obtain a generalization of Cartan’s rigidity theorem for codimension k ≤ 4. This gives a new proof of Cartan’s result that is independent of the methods of [3]. The fact that we have restricted ourselves to codimensions k ≤ 4 seems to be a technical point, and we will return to that in a while. As a simple consequence of our methods, we obtain an improvement, for codimension k ≤ 5, of Allendoerfer’s isometric rigidity theorem [1].