Weingarten变换下曲面的特征向量和特征值的一种求法

A Method for Finding Eigenvectors and Eigenvalues of Surfaces under Weingarten Transformation

本文运用高等代数中n阶矩阵特征值求解的传统方法,来求出一些正则曲面Weingarten变换下曲面的特征值。具体而言就是先求出Weingarten矩阵特征方程的全部特征根,然后依据Weingarten变换定义来求出曲面的特征向量。这里求特征向量的最大区别在于特征向量要先用给定曲面在讨论点处切平面的坐标曲线切向量基底表示,再根据曲面的第一基本量和特征向量的单位来求出相应的分量。以一些特有性质的正则曲面为例子,来阐述它们的Weingarten变换的特征值和特征向量求解方法,加深对求曲面特征值和特征向量的理解。

In this paper,by using the traditional method of solving the eigenvalues of matrix with order n in Advanced Algebra,the eigenvalues and corresponding eigenvectors of some regular surfaces under Weingarten transformation are obtained.Precisely,after obtaining all eigenvalues of characteristic equation of the Weingarten matrix,we solve the eigenvector corresponding to its eigenvalue.The biggest distinction is in that the eigenvector is expressed by the basis formed by tangent vectors of coordinate curves with respect to the tangent plane on given surface.Some examples of canonical surfaces are used to illustrate the methods for solving eigenvalues and eigenvector to their Weingarten transformations,which help us a better understanding of eigenvalue and eigenvector of surfaces.