Rigidity of CR submanifolds and analyticity of CR immersions (Q1122686)

Let M and \(\tilde M\) be real analytic CR manifolds of hypersurface type of dimension \(2n+1\) and \(2\tilde n+1\), respectively, where \(n<\tilde n\). Let f be a CR immersion of M into \(\tilde M.\) Suppose that \(\tilde M\) is strongly pseudoconvex and f is of differentiability class \({\mathcal C}^ 5\) and f is rigid at every point of M. Then f is real analytic. In particular, a \({\mathcal C}^ 5\) immersion of a real analytic CR manifold of dimension 2n-1 into the sphere \(S^{2n+1}\) is real analytic if \(n\geq 4\).