A Proof And Application Of The Index Thereom Of Atyiah-Singer

Atiyah-Singer index theorem has a important significance in the 20th century this theorem contains three important theorems in other disciplines, respectively is: the Gauss-Bonnet-Chern theorem of differential geometry, Hirzebruch theorem in topology, and algebraic geometry of Riemann Roch theory.A smooth manifold for even dimensional has a equation was established as follows:It is about the analysis and the formula on the left, but at the same time, the formula on the right is a characteristic class integral, because the characteristic class is topological meaning, so the right reflects the topological properties of the manifold itself, simply Atiyah-Singer index theorem gives a connection between topology and analysis. This Theorem has different proof methods, but by the formula, there must be proof of different Dirac operator, characteristic classes and Clifford module, in fact the definition of Dirac operator is performed in the Clifford module. The next basic concepts are presented in Atiyah-Singer index theorem are introduced in this article and we utlize the properties of the heat equation to prove it, we also demonstrated the connection between the Gauss-Bonnet-Chern thereom and the Atiyah-Singer thereom...