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

TL;DR

This paper introduces a new form to express the Gauss-Bonnet-Chern theorem, explores its topological structure, and generalizes the Morse theory formula for the Euler characteristic.

Contribution

It presents a novel one-form based on the second fundamental tensor, linking the Gauss-Bonnet-Chern form with delta-functions and extending Morse theory for Euler characteristic.

Findings

Expressed Gauss-Bonnet-Chern form using the new one-form

Linked Gauss-Bonnet-Chern density with delta-functions via phi-mapping theory

Generalized Morse theory formula for Euler characteristic

Abstract

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.