Gauss-Bonnet-Chern定理与Morse定理的内在结构
Inner structure of Gauss-Bonnet-Chern Theorem and the Morse theory
摘要：
本文在第二基本张量H^abA的基础上定义了一种新的单形H^A，Gauss-Bonnet-Chern形式可以用这种单形来表示.利用φ-映射理论，我们发现Gauss-Bonnet-Chern密度可以用δ-函数表示，并直接给出了Gauss-Bonnet-Chern定理与Hopf-Poincare定理之间的关系.详细讨论了Gauss-Bonnet-Chern定理的拓扑流及其拓扑结构。最后，推广了欧拉特征线的Morse理论公式.
英文摘要：
We define a new one form H^A based on the second fundamental tensor H^abA, the Gauss-Bonnet-Chern form can be novelly expressed with this one-form. Using the phi-mapping theory we find that the Gauss-Bonnet-Chern density can be expressed in terms of the delta-function and the relationship between the Gauss-Bonnet-Chern theorem and Hopf-Poincare theorem is given straightforwardly. The topological current of the Gauss-Bonnet-Chern theorem and its topological structure are discussed in details. At last, the Morse theory formula of the Euler characteristic is generalized.
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