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R. C. Reilly calculated the variations of functions of the mean curvatures for hypersurfaces in Euclidean space. In the present paper, using Reilly’s formulas, we derive some general integral formulas for hypersurfaces, which generalize the well-known Minkowski formulas, and then apply those formulas to obtain some characterizations of the hypersphere.

Some integral formulas for hypersurfaces and a generalization of the Hilbert-Liebmann theorem