Gauss parametrizations and rigidity aspects of submanifolds (Q1073352)

If the Gauss map \(\phi\) : \(M^ n\to S^ n\) of a hypersurface \(M^ n\subset {\mathbb{R}}^{n+1}\) has rank\(\geq 3\), then \(M^ n\) is known to be rigid. If this rank is everywhere two then \(\phi (M^ n)\) is a two- dimensional surface in \(S^ n\), the geometry of which reflects to a large extent the geometry of \(M^ n\). This relationship is used to prove many rigidity results, including the following: If a hypersurface \(M^ n\subset {\mathbb{R}}^{n+1}\) has constant mean curvature and neither is contained in a cylinder \(S^ 1\times {\mathbb{R}}^{n-1}\) nor in a hypersurface of the form \(L^ 2\times {\mathbb{R}}^{n-2}\) \((L^ 2\) of constant mean curvature in \({\mathbb{R}}^ 3)\), then \(M^ n\) is rigid.