Geodesic and curvature of piecewise flat Finsler surfaces

TL;DR

This paper investigates the properties of geodesics and curvature on piecewise flat Finsler surfaces, introducing new classifications and formulas that extend classical geometric results to this discrete Finsler setting.

Contribution

It defines Landsberg and Berwald types of piecewise flat Finsler surfaces, derives conditions for geodesic extension at vertices, and generalizes the Gauss-Bonnet formula to this context.

Findings

Explicit condition for geodesic extension at vertices

Introduction of a curvature measure related to geodesic extensions

A generalized Gauss-Bonnet formula for Landsberg type surfaces

Abstract

A piecewise flat Finsler metric on a triangulated surface $M$ is a metric whose restriction to any triangle is a flat triangle in some Minkowski space with straight edges. One of the main purposes of this work is to study the properties of geodesics on a piecewise flat Finsler surface, especially when it meets a vertex. Using the edge-crossing equation, we define two classes of piecewise flat Finsler surfaces, namely, Landsberg type and Berwald type. We deduce an explicit condition for a geodesic to be extendable at a vertex, and define the curvature which measures the \textit{amount} of such extensions. The dependence of the curvature on an incoming or outgoing tangent direction corresponds to the feature of flag curvature in Finsler geometry. When the piecewise flat Finsler surface is of Landsberg type, the curvature is only relevant to the vertex, and we prove a combinatoric Gauss-Bonnet formula which generalizes both the Gauss-Bonnet formulas for piecewise flat Riemannian manifolds and for smooth Landsberg surfaces.