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

TL;DR

This paper constructs a broad class of projectively equivalent $C^0$-Finsler manifolds on $\

Contribution

It introduces a large family of $C^0$-Finsler manifolds with explicit geodesic structures, not invariant under transformations, expanding understanding of non-smooth Finsler geometries.

Findings

Unique minimizing paths are line segments or concatenations thereof.

The manifolds are not Busemann $G$-spaces.

They lack bounded open $\\hat F$-strongly convex subsets.

Abstract

Let $M$ be a differentiable manifold and $TM$ be its tangent bundle. A $C^0$-Finsler structure on $M$ is a continuous function $F: TM \rightarrow \mathbb R$ 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.