Remarks on the Darboux transform of isothermic surfaces
Remarks on the Darboux transform of isothermic surfaces
复制标题
关于等温面达布变换的备注
DOI：
10.4171/dm/32
复制
发表时间：
1997
影响因子：
0.9
通讯作者：
F. Pedit
中科院分区：
文献类型：
--
作者：
U. Hertrich;F. Pedit
本文研究了欧氏空间中等温曲面的达布变换和克里斯托-埃尔变换。使用四元数微积分，我们推导出一个Riccati型方程的特征所有达布变换的一个给定的等温表面。常平均曲率曲面是等温曲面中的一个特殊曲面：它们的平行常平均曲率曲面同时是Christo-el变换和Darboux变换。我们证明|作为关于极小曲面的极小Darboux变换的比安奇定理的推广|欧氏空间中常平均曲率曲面允许1 - 3达布变换为常平均曲率曲面。指出了这些达布变换与B之间的关系球面的Acklund变换1991年数学科目分类：（小学）53 A10、（中学）53 A50、53 C42。
We study Darboux and Christo el transforms of isothermic surfaces in Euclidean space. Using quaternionic calculus we derive a Riccati type equation which characterizes all Darboux transforms of a given isothermic surface. Surfaces of constant mean curvature turn out to be special among all isothermic surfaces: their parallel surfaces of constant mean curvature are Christo el and Darboux transforms at the same time. We prove | as a generalization of Bianchi's theorem on minimal Darboux transforms of minimal surfaces | that constant mean curvature surfaces in Euclidean space allow 1 3 Darboux transforms into surfaces of constant mean curvature. We indicate the relation between these Darboux transforms and Backlund transforms of spherical surfaces. 1991 Mathematics Subject Classi cation: (Primary) 53A10, (Secondary) 53A50, 53C42.