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

Abstract

A $C^0$-Finsler structure on a differentiable manifold is a continuous real valued function defined on its tangent bundle 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 $(\hat M,\hat F)$, where $\hat M$ is diffeomorphic to the Euclidean plane. The 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.