Inner structure of Gauss-Bonnet-Chern Theorem and the Morse theory

Details Inner structure of Gauss-Bonnet-Chern Theorem and the Morse theory Inner structure of Gauss-Bonnet-Chern Theorem and the Morse theory

2002-12-19

arxiv.org/abs/math-ph/0212055v1

Mod.Phys.Lett. A16 (2001) 2483

Physics

Mathematical Physics

10 pages

Scientific paper

10.1142/S0217732301006004

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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