# Congruence theorems for compact hypersurfaces of a riemannian manifold

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## Summary

Let Mm,$$\bar M$$ m be two m-dimensional compact oriented hypersurfaces of class C3 immersed in a Riemannian manifold Rm+1 of constant sectional curvature. Suppose that Rm+1 admits a one-parameter continuous group G of conformal transformations satisfying a certain condition (which holds automatically when G is a group of isometric transformations). Suppose further that there is a1 − 1 transformation Tτ ∈ G between Mm and$$\bar M$$ m such that$$\bar P = T_{\tau (P)} P$$ for each P ∈ Mm and each$$\bar P \in \bar M$$ m. If the r-th mean curvature for any r, 1 ⩽ r ⩽ m, of Mm at each point P ∈ Mm is equal to that of$$\bar M$$ m at the corresponding point$$\bar P = T_{\tau (P)} P$$, together with other conditions, then Mm and$$\bar M$$ m are congruent mod G. This is a generalization of a joint theorem ofH. Hopf andY. Katsurada  in which G is a group of isometric transformations.

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## Author information

Entrata in Redazione il 13 Giugno 1975.

The first author was partially supported by the National Science Foundation grant GP-33944.

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Reprints and Permissions

Hsiung, C., Lo, T.P. Congruence theorems for compact hypersurfaces of a riemannian manifold. Annali di Matematica 109, 289–304 (1976). https://doi.org/10.1007/BF02416965