常曲率黎曼流形容有正交全脐超曲面族的某些特征

本文讨论了当黎曼流形容有二族彼此正交的全脐超曲面时,这些超曲面应满足怎样的条件该流形才是常曲率的。所得结果完善了胡和生教授早先在文中所给出的一个结论。

On Some Characteristics of the Constant Curvature Riemannin Manifold Admitting Orthogonal Family of Totally Umbilical Hypersurfaces

In the present peper we obtain the following theorems. Theorem 1. Suppose that an n(>3) dimensional Riemannian manifold M admits two mutually orthogonal families of totally umbilical hypersurfaces among which one is Einsteinian:En-1's and the other is of constant curvature: Sn-1's. If either En-1 or Sn-1 has Constant mean curvature( 0),then the M must be of constant curvature. Theorem 2. Suppose that an n(>3) dimensional Riemannian manifold M admits two mutually orthogonal families of totally umbilical hypersurfaces among which one is Einsteinian and the other is of constant curvature. If the scalar curvature K of M and U curvatures of these hypersurfaces satisfy the condition k=1/2n(n-1)(U1+U2) then the M certainly is of constant curvature. These theorms characterize the Riemannian manifold of constant curvature.