微分流形几何学与拓扑学的历史研究

Differential manifold is one of the most important concepts and theories, the geometry and topology developed on the basis of which belong to the category of structural mathematics, including several disciplines and branches, such as Riemannian geometry, algebraic topology, complex function, algebraic geometry and differential topology, are in the mainstream position of modern mathematics. This project mainly focuses on the history of geometry and topology of differential manifold, as well as Riemannian surface and Lie group. The main research topics are as follows: the development of curvature and connection from the second half of the 19th century to the 1920s, the evolution of homology and cohomology from the late 19th century to the 1930s, the dissemination of Riemannian surface from the latter half of the 19th century to the beginning of the 20th century, and the formation of global Lie group, as well as the biographies of Levi-Civita, Alexander, Hopf, Veblen and other mathematicians. By the implement of this project, it will help us straighten the history of geometry and topology of differential manifold, Riemannian surface and Lie group, clarify their roles and effects in the development of geometry and topology, refine the historical research of geometry and topology, and deepen the understanding of the unity of mathematics.

微分流形是现代数学最重要的概念和理论之一，以其为基础发展起来的几何学与拓扑学属于结构数学的范畴，涉及到学科有黎曼几何、代数拓扑、复变函数、代数几何与微分拓扑等分支，在现代数学中处于主流的位置。本项目主要对微分流形的几何学与拓扑学以及与之有密切关联的黎曼曲面与李群进行历史研究，主要研究问题如下：19世纪下半叶到20世纪20年代曲率、联络等概念和理论的发展，19世纪末到20世纪30年代同调、上同调等概念与理论的发展，黎曼曲面在19世纪下半叶到20世纪初的传播，整体李群的概念和理论是如何形成的，以及列维-奇维塔、亚历山大、霍普夫与维布伦等相关数学家的人物评传。通过本项目的实施，将有助于厘清微分流形的几何学与拓扑学以及与之相关的黎曼曲面和李群的发展历史，明晰它们在几何与拓扑学发展过程中的作用和影响，深入和细化几何学与拓扑学的历史研究，加深人们对数学统一性的认识。

本项目主要研究微分流形的几何学与拓扑学，以及与之密切相关的黎曼曲面和李群理论的历史。首先，项目梳理了19世纪中叶以来，微分流形几何学与拓扑学的发展脉络，总结了维布伦对微分流形公理化的贡献，怀尔德对流形拓扑学的贡献，惠特尼对微分拓扑的贡献，吴文俊早期与惠特尼在拓扑学方面的学术渊源，以及廖山涛对代数拓扑的贡献。其次，项目重点研究了克莱因对黎曼曲面的传播，翻译了克莱因1882年的著作《关于代数函数及其积分的黎曼理论》；对外尔在微分流形的基础上奠基整体李群的工作进行了考察，剖析了这一工作对后世的重要影响。最后，对维布伦、惠特尼、廖山涛等数学家进行了详细的传记研究，揭示了他们的成才路径、师承关系、学术贡献和数学思想。本项目深化了以微分流形为中心的几何学与拓扑学的历史研究，对于理解现代数学的统一性提供了历史注解与案例。