球空间 S^(n+p)(c)中的紧致极小子流形

THE COMPACT MINIMAL SUBMANIFOLD IN SPHERE S^(n+p)(c)

设 M^n 是常曲率空间 S^(n+p)(c)的紧致极小子流形,设 K 和 Q 分别是 M^n 上每点各方向截面曲率和 Ricci 曲率的下确界,R 是 M^n 的数量曲率,本文利用 M^n 的内在量 KQ和 R,给出球空间中紧致极小子流形是全测地子流形的六个充分条件。

Let M^n be an n—dimensional compact minimal submanifold in S^(n+p)(c)with constant curvature c(>0).Let K and Q be the infimum of the Sectionalcurvature and Ricci curvature of M^n respectively.Let R be the Scalarcurvature of M^n.In this paper,we obtain the following result:Let M^n be acompact minimal submanifold in S^(n+p)(c).if one of the following conditions issatisfied:(i)Q≥(n-1)c-n/2K,(ii)R>(n^2+2n-4)c-4Q,(iii)R>n(n-1)c-(np)/(p-1) c (p>1) (iv)K>1/2[n(n-1)c-R],(v)R>(n^2+n-4)c-2Q,(vi)R<n(n-1)c-np[(2n-3)c-2Q],then M^n is totally geodesic submanifold.