S^4(1)中具有两个相等的非零常数主曲率的完备超曲面

COMPLETE HYPERSURFACES IN S^4 (1) WITH TWO EQUAL CONSTANT PRINCIPAL CURVATURE

设 M 是连通的、可定向的、完备的3维 C~∞黎曼流形,C:M→S^4(1)是从 M 列4维单位球面 S^4(1)中的等距浸入.主曲率 h_1,h_2,h_3满足 h_1=h_2=R(常数).本文证明了:浸入或者是全脐的,或者是无脐点的;若浸入是全脐的.或无脐点且 h_3为常数,则 M 可完全确定:若 h_3不是常数,则 M 微分同胚于 E^4中环准超环面.

Let M be a connected orientable complete 3-dimensional C~∞ Riemannian manifold,τ:M—S^4(1)an isometric immersion of M into a 4-dimensional unit sphere S^4(1),such that principal curvature h_1,h_2,h_3have h_1=h_2=R(constant).In this pa- per,we prove that τ is either totally umbilical or else umbilic free;and that if τ is totally umbilical,then τ(M)is congruent to a 3—dimensional sphere S^3(1/(1+R^2)^(1/2))with radii 1/(1+R^2)^(1/2),but if τ is umbilic free,then either h_3=-1/R,and τ(M) is comgruent to S^2(1/(1+R^2)^(1/2))×S^1(R/(1+R^2)^(1/2)),or h_3≠constant,in addition,τis an imbedding,then τ(M) diffeomorphic to a standard hypertori in E^4.