欧氏空间中给定主曲率函数的旋转超曲面

Rotational hypersurface about given principal curvature function in euclidean space

经典微分几何研究三维欧氏空间中曲线曲面理论,其最具有特色的研究是主曲率函数满足某些关系的魏因加吞曲面。一般地说,这种曲面的研究归结为某个二阶椭圆型偏微分方程的求解。由于求解这种偏微分方程相当困难,许多微分几何学家利用曲面某些特殊的性质,将偏微分方程的求解转化为常微分方程或方程组的求解。在此基础上利用超曲面的旋转对称性,给出了欧氏空间Rn+1中给定主曲率函数旋转超曲面的位置向量场后,计算出这种超曲面的主曲率,通过求解相应的常微分方程组,证明了这类超曲面的存在性。

In the classical differential geometry which deals with the theory of curves and surfaces of three dimensional Euclidean space, the most distinctive study is the Weingarten surface. Generally, the research of this surface can be reduced to the solving of a partial differential equation. This thesis gives the position vector field of the rotational hypersurface in Euclidean space. After computing the principal curvatures and solving the differential equation system, the thesis proves the existence of the rotational hypersurface about given principal curvature function in Euclidean space.