Congruence theorems for compact hypersurfaces of a riemannian manifold

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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 [5] in which G is a group of isometric transformations.


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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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Hsiung, C., Lo, T.P. Congruence theorems for compact hypersurfaces of a riemannian manifold. Annali di Matematica 109, 289–304 (1976).

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  • Riemannian Manifold
  • Sectional Curvature
  • Conformal Transformation
  • Continuous Group
  • Constant Sectional Curvature