Congruence theorems for compact hypersurfaces of a riemannian manifold

Summary Let M^m, ^m be two m-dimensional compact oriented hypersurfaces of class C³ immersed in a Riemannian manifold R^m+1 of constant sectional curvature. Suppose that R^m+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 T_τ ∈ G between M^m and ^m such that for each P ∈ M^m and each ^m. If the r-th mean curvature for any r, 1 ⩽ r ⩽ m, of M^m at each point P ∈ M^m is equal to that of ^m at the corresponding point , together with other conditions, then M^m and ^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. Entrata in Redazione il 13 Giugno 1975. The first author was partially supported by the National Science Foundation grant GP-33944.