Compact hypersurfaces with constant scalar curvature and a congruence theorem (Q1099419)

Using integral formulas the following main results are shown which generalize classical congruence theorems: a) The sphere is the only compact hypersurface with constant scalar curvature embedded in the Euclidean space. b) Let \(\psi\) : \(M^ n\to {\mathbb{R}}^{n+1}\) be a compact hypersurface embedded in the Euclidean \((n+1)\)-space. Let \(\psi\) ': \(M^ n\to {\mathbb{R}}^{n+m}\) be an isometric immersion. If the mean curvature vector H' of \(\psi\) ' satisfies \(| H'| \leq H\) everywhere (H denoting the mean curvature of \(\psi)\), then \(\psi\) ' differs from \(\psi\) by a rigid motion.