A proof of the Chern-Gauss-Bonnet theorem for indefinite signature metrics using analytic continuation

A generalization of the Gauss-Bonnet and Hopf-Poincaré theorems. Part II

This paper is a continuation of the paper F. A. Arias and M. Malakhaltsev "A generalization of the Gauss-Bonnet and Hopf-Poincaré theorems", ArXiv:1510.01395 [MathDG] 5 Oct 2015. Let $π: E \to M$ be a locally trivial fiber bundle over a two-dimensional manifold $M$, and $Σ\subset M$ be a discrete subset. A subset $Q \subset E$ is called an $n$-sheeted branched section of the bundle $π$ if $Q' = π^{-1}(M \setminus Σ) \cap Q$ is a $n$-sheeted covering of $M \setminus Σ$. The set $Σ$ is called the singularity set of the branched section $Q$. We define the index of a singularity point of a branche…

曲率上有界的二维度量空间 II

Two-dimensional metric spaces with curvature bounded above II

本文延续前期工作，研究曲率上有界的二维局部紧致测地完备度量空间的整体结构，通过多面体空间进行Lipschitz同伦逼近，定义曲率测度并建立Gauss-Bonnet定理，同时给出空间刻画。