2016/08/21 by Ming Xu, Xu, Ming, Shaoqiang Deng +1
Physics and Astronomy · #52B70 #53C60 #Advanced Differential Geometry Research #Combinatorics (math.CO) #Differential Geometry (math.DG) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.1608.05896
openalex publication_date 2016/08/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 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.