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Compact hypersurfaces with constant scalar curvature and a congruence theorem. (English) Zbl 0638.53051
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.
Reviewer: Bernd Wegner

MSC:
53C40 Global submanifolds
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