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

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.