Mollifier smoothing of $C^0$-Finsler structures

Abstract

A C 0 -Finsler structure is a continuous function F : T M → [0, ∞) defined on the tangent bundle of a differentiable manifold M such that its restriction to each tangent space is an asymmetric norm. We use the convolution of F with the standard mollifier in order to construct a mollifier smoothing of F , which is a one parameter family of Finsler structures Fε (of class C ∞ on T M \0) that converges uniformly to F on compact subsets of T M . We prove that when F is a Finsler structure, then the Chern connection, the Cartan connection, the Hashiguchi connection, the Berwald connection and the flag curvature of Fε converges uniformly on compact subsets to the corresponding objects of F . As an application of this mollifier smoothing, we study examples of two-dimensional piecewise smooth Riemannian manifolds with nonzero total curvature on a line segment. We also indicate how to extend this study to the correspondent piecewise smooth Finsler manifolds.