A large family of projectively equivalent $C^0$-Finsler manifolds

Abstract

Let $M$ be a differentiable manifold and $TM$ be its tangent bundle. A $C^0$-Finsler structure on $M$ is a continuous function $F:TM\to \mathbb{R}$ such that its restriction to each tangent space is a norm. In this work we present a large family of projectively equivalent $C^0$-Finsler manifolds $(\hat{M}\cong {\mathbb{R}}^2,\hat{F})$. Their structures $\hat{F}$ don't have partial derivatives and they aren't invariant by any transformation group of $\hat{M}$. For every $p,q\in (\hat{M},\hat{F})$, we determine the unique minimizing path connecting $p$ and $q$. They are line segments parallel to the vectors $(\sqrt{3}/2,1/2)$, $(0,1)$ or $(-\sqrt{3}/2,1/2)$, or else a concatenation of two of these line segments. Moreover $(\hat{M},\hat{F})$ aren't Busemann $G$-spaces and they don't admit any bounded open $\hat{F}$-strongly convex subsets. Other geodesic properties of $(\hat{M},\hat{F})$ are also studied.

arXiv comment26 pages, accepted in Tohoku Mathematical Journal