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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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