高斯-博内-陈定理的历史发展及其意义

The Historical Development and the Meaning of Gauss-Bonnet-Chern Theorem

高斯-博内定理是大范围微分几何学的一个经典定理,它建立了黎曼流形的局部性质和整体性质之间的联系,因而被认为是曲面微分几何学中最深刻的定理.通过考察高斯-博内-陈定理的历史发展,指出高斯-博内-陈定理在黎曼流形、微分流形以及拓扑流形上的表现形式,以此阐明高斯-博内-陈定理与现代数学的深刻联系及其意义.

Gauss-Bonnet theorem,which established a link between the local nature of Riemannian manifold and the overall’s,is a classic theorem of differential geometry in the large,and considered to be the most profound theorem in differential geometry.The historical development of the Gauss-Bonnet-Chern theorem are examined,the manifestations of the Gauss-Bonnet-Chern theorem in Riemannian manifolds,differential manifolds,and topology manifolds are dis-cussed,and the deep relations and meaning between the Gauss-Bonnet-Chern theorem and the modern mathematics are clarified as well.