球面上的两个公式及其运用

Two Formulas on the Unit Sphere and Their Applications

得到了欧氏空间中,单位球面上坐标函数关于某一特定标架场的协变微分的两个等式.它们有非常重要的应用.首先它们可以用来求出球面上作用在形式上的拉普拉斯算子的谱及重数,而在文献中这些结果并不清楚.其次还可以用这两个等式来计算欧氏空间中某些特殊类型凸超曲面的各阶平均曲率.

We obtain two formulas about the covariant derivative of the coordinate tunctions on me unit sphere in Eucdidean space. As applications we get the correct form of the eigenforms and eigenvalues of Laplacian acting on differential forms in the sphere. We also get a simple calculation of Weingarten curvature of a special convex hypersurface in Euedidean space. Our method can be generalized to the non-rotational case.