再论F.克莱因的《埃尔朗根纲领》

Rethinking of Felix Klein's Erlangen Program

埃尔朗根纲领是数学史上最重要的文献之一,其核心思想是群统一几何。除此之外,克莱因在埃尔朗根纲领中还论述了二元型与几何学的对应关系、李球几何与切触变换等内容,并试图将总结的几何学研究推广到流形上。由于克莱因的流形都是齐性空间,所以他的几何学分类中没有包含黎曼几何。

Erlangen Program is most important in the history of mathematics, whose core idea ~s the tmification of geometry by means of transformation group. In the Program, Klein also discusses the connection between binary form and geometry, Lie sphere geometry, contact transformation, and tries to extend the summarized geometry researches to manifold. As Klein＇s manifolds are homogeneous spaces, his classification does not include Riemannian geometry.