On the Curvature of Regge Metrics

TL;DR

This paper develops a distributional curvature concept for Regge metrics using moving frame techniques, proving its consistency with existing notions and establishing the Gauss-Bonnet theorem in two dimensions.

Contribution

It introduces a new distributional curvature for Regge metrics that satisfies Cartan equations and gauge laws, linking it to existing curvature concepts and proving a fundamental theorem in 2D.

Findings

Distributional curvature for Regge metrics is equivalent to existing notions.

The curvature satisfies weak Cartan structure equations and gauge transformations.

Gauss-Bonnet theorem is proven for 2D Regge metrics.

Abstract

We use moving frame techniques to derive a notion of curvature for a class of piecewise-smooth Riemannian metrics called Regge metrics, showing that it is a measure that simultaneously satisfies the (weak) Cartan structure equations and the appropriate gauge transformation law. It turns out that this distributional curvature is equivalent to existing notions of densitized distributional curvature. We also investigate more closely the n = 2 case, where we prove the Gauss-Bonnet theorem for Regge metrics.