A generalization of the Gauss-Bonnet and Hopf-Poincaré theorems

2015/10/05 by Arias, F. A., Malakhaltsev, M.
#53C10 #55S35 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1510.01395

Abstract

We consider a locally trivial fiber bundle π: E → M over a compact oriented two-dimensional manifold M, and a section s of this bundle defined over M ∖ Σ, where Σ is a discrete subset of M. We call the set Σ the set of singularities of the section s : M ∖ Σ→ E. We assume that the behavior of the section s at the singularities is controlled in the following way: s(M ∖ Σ) coincides with the interior part of a surface S ⊂ E with boundary ∂ S, and ∂ S is π^-1(Σ). For such sections s we define an index of s at a point of Σ, which generalizes in the natural way the index of zero of a vector field, and then prove that the sum of this indices at the points of Σ can be expressed as integral over S of a 2-form constructed via a connection in E. Then we show that the classical Hopf-Poincaré-Gauss-Bonnet formula is a partial case of our result, and consider some other applications. Keywords: singularity of section, index of singular point, curvature, projective bundle, G-structure

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