关于局部对称共形平坦黎曼流形上极小子流形的内蕴刚性

The Minimal Submanifolds in a Locally Symmetric Conformally Flat Riemannian Manifold

设Ｍ是局部对称共形平坦黎曼流形Ｎｎ＋ｐ的紧致极小子流形，Ｋｃ和Ｑ分别是Ｍ上每点截面曲率和Ｒｉｃｃｉ曲率的下确界，Ｒ是Ｍ的数量曲率，本文利用三种内蕴量Ｋｃ，Ｑ和Ｒ所满足的适当关系。

In this paper we prove the following theorems: Theorem 1 If M be a compact minimal submanifold of a Locally symmetric and conforally fiat riemannian manifold N n+p .If: (i) K c>-p-1pnR+1n+p-2 ×2(p-1)(n-1)+ppT c-t c-(p-1)(n-1)p(n+p-1)K , or (ii) K c>-12R+1n+p-2(n 2-n+1)T c-t c-n(n-1)2(n+p-1)K. then M is totally geodesic. Theorem 2 If: n≥4 (i) R>-2Q+1n+p-2(2n 2+2n-6)T c-2t c-n 2+n-4n+p-1K , or (ii) R<2npQ+nn+p-2 ×(2n-2)t c-p(4n-6)T c+2np-3p-n+1n+p-1K. then M is totally geodesic submanifold.