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Inner structure of Gauss–Bonnet–Chern theorem and the Morse theory. (English) Zbl 1138.37303

Summary: We define a new one-form \(H^A\) based on the second fundamental tensor \(H_{ab \overline{A}}\) , the Gauss–Bonnet–Chern form can be novelly expressed with this one-form. Using the \(\varphi\)-mapping theory we find that the Gauss–Bonnet–Chern density can be expressed in terms of the \(\delta\)-function \(\delta (\varphi)\) and the relationship between the Gauss–Bonnet–Chern theorem and Hopf–Poincaré 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.

MSC:

37B30 Index theory for dynamical systems, Morse-Conley indices
53C17 Sub-Riemannian geometry
57R20 Characteristic classes and numbers in differential topology
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
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