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 ( M , F ), where M is diffeomorphic to the Euclidean plane. The structures F don't have partial derivatives and they aren't invariant by any transformation group of M . For every p, q ∈ ( M , F ), we determine the unique minimizing path connecting p and q. They are line segments parallel to the vectors ( √ 3/2, 1/2), (0, 1) or (− √ 3/2, 1/2), or else a concatenation of two of these line segments. Moreover ( M , F ) aren't Busemann G-spaces and they don't admit any bounded open F -strongly convex subsets. Other geodesic properties of ( M , F ) are also studied.

Mollifier smoothing of $C^0$-Finsler structures

“…Moreover, if x ∈ R 2 and y ∈ T x R 2 , then there exist infinite geodesics γ such that γ(0) = x and γ ′ (0) = y. Another example is in [12], where the author creates a large family of projectively equivalent C 0 -Finsler manifolds such that "most of" their minimizing paths are concatenation of two line segments. The elements of this family can be obtained by continuous deformations of a fixed C 0 -Finsler manifold.…”

“…Remark 1.1. In [4] and [12], the first author and his collaborators used the term C 0 -Finsler structure for a function F : T M → R such that F (x, •) : T x M → R is a norm. In this work the term C 0 -Finsler structure is modified in order to work with non-symmetric structures.…”