黎曼论几何学基础之假设

Riemann's analysis on the Hypotheses of the Foundation of Geometry

《论奠定几何学基础之假设》是黎曼在哥廷根大学所做的讲师试讲,是现代微分几何学的奠基性文献,具有重要的数学意义与哲学意义。黎曼在文中提出流形、度量、曲率等概念,用以澄清关于空间本质和几何学的基本构建的问题,其洞见直接影响了之后半个多世纪的数学家,并带来了空间观念的变革。在空间哲学、非欧几何学、微分几何学三重背景下,考察黎曼演讲所要解决的问题,分析黎曼演讲的内容,总结该演讲的历史贡献。

Riemann＇ s probationary lecture at the University of Gottingen ＂On the hypotheses which lie at the basis of geometry＂ lays a foundation for modem differential geometry, which has great mathematical and philosophical meaning. Riemann put forward the concept of manifold, metric, and curvature in this thesis to clarify the basic problems about the nature of space and the basic founding of geometry, which directly influenced the mathematicians of more than half a century and brought about the change of the concept of space. Under the triple background of the philosophical discussion about space, non-Euclidean geometry and differential geometry, this article examines the problems that Riemann tries to solve, interpret Riemann＇ s lecture, and analyze its contribution in history.