Transformations of surfaces
Transformations of surfaces
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表面的变换
DOI：
10.2307/3605517
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发表时间：
1934
期刊：
影响因子：
--
通讯作者：
L. Eisenhart
中科院分区：
文献类型：
--
作者：
L. Eisenhart
在以前的文献中，我们发展了一个关于具有相等点不变量的共轭系统到同类系统的变换K的理论。随后，我们考虑的情况下，连接相应的两个表面上的点的关系的变换K的线形成一个正常的同余。这些曲面是一种特殊的曲面，称为曲面C。我们已经证明了存在K型变换，记作Km I，它把曲面C变换成曲面C。与这些法同余正交的曲面S就是Demoulin所讨论的曲面Q。我们证明了存在S到表面Si的变换Am，使得S和表面Si包围一个双参数球面族，并且在S和Si之间的对应中，因此建立的曲率线对应。在本回忆录中，我们扩展了我们对这些变换Am的研究，特别是将结果应用于某些类型的曲面Q。变换Am属于一般类的变换Ribaucour，并在第一部分的文件，我们把方程的形式所拥有的所有变换Ribaucour。Guichard发现了一类具有下列特征性质的曲面：如果S是这样的曲面，则存在一个关联曲面Si，使得两个曲面上的曲率线具有相同的球面表示，并且相应曲面的主曲率半径P1，P2和p ′，p2满足关系式
In a former memoirt we developed a theory of transformations K of conjugate systems with equal point invariants into systems of the same kind. Subsequently$ we considered the case in which the lines joining corresponding points on two surfaces in the relation of a transformation K form a normal congruence. These surfaces, being of a particular kind, are called surfaces C. We have shown that there are transformations of the K type, denoted by Km I which transform a surface C into surfaces C. The surfaces S, orthogonal to these normal congruences, are the surfaces Q discussed by Demoulin.? We showed that there exist transformations Am of S into surfaces Si, such that S and a surface Si envelope a two-parameter family of spheres, and in the correspondence between S and Si thus established lines of curvature correspond. In the present memoir we extend our investigations concerning these transformations Am , and more particularly apply the results to certain types of surfaces Q. Transformations Am belong to the general class of transformations of Ribaucour, and in the first part of the paper we put the equations in a form possessed by all transformations of Ribaucour. Guichard discoveredfl a class of surfaces possessing the following characteristic property: If S is such a surface, there exists an associate surface Si such that the lines of curvature on the two surfaces have the same spherical representation, and the principal radii of curvature P1, P2 and p', p2 of the respective surfaces are in the relation