球面中的极小浸入子流形

Submanifolds Minimally Immersed in a Sphere

通过对第二基本形式的长度平方‖h‖~2 的取值的研究,证明了 ‖h‖~2 的值仅依赖于 Ricci 曲率在这个浸入的梯度方向的值,应用此结论证明了:如果那么 M^n 是全测地的,或 M^n 是 Veronese 曲面,或 M^n 是 S^(n+1)(1)中的超曲面S^k((k/n)^(1/2))×S^(n-k)(((n-k)/n)^(1/2))。其次研究了法曲率平坦的子流形。

Proves that the value of ||h||~2, the square of length of the second fundamentalform, only depends on that of the Ricci curvature immersed in the directionof gradient, based on a discussion made on the range of values of ||h||~2. As aresult, it is proved that if the Ricci curvature of M^n satisfies Ric (?x^A, Vx^A)≥[(n--1)--1/(2--1/p)]||x^A||~2, then M^n is totally geodesic, a Veronese surface or ahypersurface S^k(k/n)^(1/2)×S^(n-k)((n-k)/n)^(1/2)(0<k<n) in S^(n+1)(1).The submanifoldwith flat normal curvature is discussed as well.