Two-dimensional metric spaces with curvature bounded above II

TL;DR

This paper explores the global structure of two-dimensional metric spaces with curvature bounds, introducing curvature measures, establishing a Gauss-Bonnet theorem, and providing characterizations, extending previous local results.

Contribution

It introduces a framework for curvature measures and characterizes two-dimensional spaces with curvature bounds, extending local results to global structure analysis.

Findings

Lipschitz homotopy approximations by polyhedral spaces

Definition of curvature measures via convergence

Gauss-Bonnet theorem for these spaces

Abstract

As a continuation of \cite{NSY:local}, we mainly discuss the global structure of two-dimensional locally compact geodesically complete metric spaces with curvature bounded above. We first obtain the result on the Lipschitz homotopy approximations of such spaces by polyhedral spaces. We define the curvature measures on our spaces making use of the convergence of the curvature measures, and establish Gauss-Bonnet Theorem. We also give a characterization of such spaces.