球空间中具有半平行第二基本形式的超曲面中的一个不等式

An Inequality for Hypersurfaces with Semi-Parallel Second Fundamental Form in the Sphere

摘要: 本文研究了球空间$S^{n+1}\left(n\ge 3\right)$ 中具有Möbius第二基本形式半平行的超曲面$M^n$ 上的Möbius第二基本形式$B$ 的二阶协变导数的分量$B_{ij,kl}$ 的极小模张量，并且得到了关于Möbius第二基本形式$B$ 的二阶协变导数模平方的两个等价的不等式：${\Vert {\nabla }^{2}B\Vert }^2\ge \frac{n\left(5n-4\right)}{n+2}{\Vert \nabla \Phi \Vert }^2-\frac{3n^2}{\left(n+2\right)\left(n+4\right)}{\Vert \delta \Phi \Vert }^2$ 以及 ${\Vert {\nabla }^{2}B\Vert }^2\ge \frac{n\left(n-1\right)}{n+2}{\Vert \nabla \Phi \Vert }^2+\frac{3n}{\left(n+2\right)\left(n+4\right)}{\Vert \Delta B\Vert }^2$ 。

Abstract: This paper studies the minimal norm tensor of the components $B_{ij,kl}$ of the second covariant derivative of the Möbius second fundamental form $B$ on hypersurfaces $M^n$ with semi-parallel Möbius second fundamental form in the sphere $S^{n+1}\left(n\ge 3\right)$ . Thereby establishing two inequalities for the squared norm of the second covariant derivative of $B$ : 1) ${\Vert {\nabla }^{2}B\Vert }^2\ge \frac{n\left(5n-4\right)}{n+2}{\Vert \nabla \Phi \Vert }^2-\frac{3n^2}{\left(n+2\right)\left(n+4\right)}{\Vert \delta \Phi \Vert }^2$ , 2) ${\Vert {\nabla }^{2}B\Vert }^2\ge \frac{n\left(n-1\right)}{n+2}{\Vert \nabla \Phi \Vert }^2+\frac{3n}{\left(n+2\right)\left(n+4\right)}{\Vert \Delta B\Vert }^2$ .