局部对称共形平坦黎曼流形中带有平坦法丛的子流形

SUBMANIFOLDS WITH FLAT NORMAL BOUNDLE IN LOCALLY SYMMETRIC CONFORMAL FLAT RIEMANNIAN MANIFOLD

设M^(n+p)是n+p维共形平坦黎曼流形,且它的黎曼张量R_(tjkl)之共变导微▽R_(tjkl)=0,则称M^(n+p)为局部对称共形平坦黎曼流形。 本文证得:若V^n(n≥2)是局部对称共形平坦黎曼流形M^(n+p)的n维紧致无边子流形,它具有平坦法丛,若V^n在任一点上的截面曲率均大于T_c-t_c/2(n+p-2),这里T_c、t_c分别是M^(n+p)的Ricci曲率在该点的上、下确界,则V^n一定是M^(n+p)的n+1维全测地子流形M^(n+1)之超曲面。

In this paper, We obtain main results as follows:Theorem Let M^(n+p) be an (n+p)-dimensional locally symmetric conformal flat manifold and V^n(n≥2) be an n-dimensional compact submanifold (without boundary) with flat normal boundle in M^(n+p). If the sectional curvature of any point in V^n is greater than (T_o-t_o)/(2(n+p-2)),in which T_o、t_o are supremum and infimum of Ricci curvature of M^(n+p) at same point,then V^n isa hypersurface of (n+1)-dimensional totally geodesic submanifold M^(n+1) in M^(n+p).