常曲率空间与全脐点超曲面

<正> 最近,胡和生证明了如下的命题:如果黎曼空间 V_(n+1)容有三系相互直交的常曲率全测地超曲面,那末 V_(n+1)是常曲率的,而且这些超曲面的曲率都相等.本文的目的是把这里全测地的条件换成较广泛的全脐点条件而证明同一结果.设 V_(+1)的基本张量是 α_(αβ)(α,β=1,…,n+1),而且超曲面V_n~((1))的方程是

ON SPACES OF CONSTANT CURVATURE AND THE TOTALLY UMBILICAL HYPERSURFACES

The main purpose of the present paper is to demonstrate the following:THEOREM.If a Riemannian space V_n(n≥4)contains 3 systems ofmutually orthogonal totally umbilical hypersurfaces of constant curvature,thenV_n is a space of constant curvature and all these hypersurfaces have the sameinvariant(?)where K_o and b are the Riemannian curvature and mean curvature of one ofthese hypersurfaces respectively,and these invariants are equal to the constantcurvature of V_n.