A generalization of the Gauss-Bonnet and Hopf-Poincar\'e theorems. Part II

TL;DR

This paper extends classical theorems like Gauss-Bonnet and Hopf-Poincaré to branched sections of fiber bundles over surfaces, introducing indices for singularities and a resolution method, with applications to differential equations.

Contribution

It generalizes the Gauss-Bonnet and Hopf-Poincaré theorems to branched sections of fiber bundles, including singularity analysis and resolution techniques.

Findings

Defined the index of singularities in branched sections.

Provided examples for projective tangent bundles.

Proved an analog of classical theorems for resolved branched sections.

Abstract

This paper is a continuation of the paper F. A. Arias and M. Malakhaltsev "A generalization of the Gauss-Bonnet and Hopf-Poincar\'e theorems", ArXiv:1510.01395 [MathDG] 5 Oct 2015. Let $\pi : E \to M$ be a locally trivial fiber bundle over a two-dimensional manifold $M$, and $\Sigma \subset M$ be a discrete subset. A subset $Q \subset E$ is called an $n$-sheeted branched section of the bundle $\pi$ if $Q' = \pi^{-1}(M \setminus \Sigma) \cap Q$ is a $n$-sheeted covering of $M \setminus \Sigma$. The set $\Sigma$ is called the singularity set of the branched section $Q$. We define the index of a singularity point of a branched section, and give examples of its calculation, in particular for branched sections of the projective tangent bundle of $M$ determined by binary differential equations. Also we define a resolution of singularities of a branched section, and prove an analog of Hopf-Poincar\'e-Gauss-Bonnet theorem for the branched sections admitting a resolution.