Mollifier smoothing of $C^0$-Finsler structures

Abstract

A $C^0$-Finsler structure is a continuous function $F:TM\to [0,\infty )$ 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 $F_{\epsilon }$ (of class $C^{\infty }$ on $TM\backslash 0$) that converges uniformly to $F$ on compact subsets of $TM$. 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 $F_{\epsilon }$ 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.

44 pages

References in corpus (1)

• Sequence of induced Hausdorff metrics on Lie groups

Cited by in corpus (2)

• Geodesic fields for Pontryagin type $C^0$-Finsler manifolds
• Extremals on Lie groups with asymmetric polyhedral Finsler structures