A functional characterization of isometries between non-reversible Finsler manifolds

TL;DR

This paper characterizes isometries between non-reversible Finsler manifolds using a novel asymmetric function space framework, extending classical results to asymmetric geometric structures.

Contribution

It introduces an asymmetric function space approach to describe isometries in non-reversible Finsler manifolds, generalizing previous symmetric cases.

Findings

Established a functional characterization of isometries for non-reversible Finsler manifolds.

Modified the algebra of Lipschitz functions to include semi-Lipschitz functions.

Extended the Myers-Nakai Theorem to asymmetric Finsler geometries.

Abstract

We provide a functional characterization of isometries between non-reversible Finsler manifolds, in the form of a generalization of the Myers-Nakai Theorem for Riemannian manifolds. We show that, since non-reversible Finsler manifolds are a fundamentally asymmetric object, such a result can not be obtained by means of a symmetric function space, and we define the appropriate asymmetric structure needed to describe all possible isometries between this class of manifolds. The result is based on the ideas used in a previous generalization for reversible Finsler manifolds proved in \cite{GJR-13}, in which the normed algebra of $C^1$-smooth Lipschitz functions is used. To reflect the quasi-metric structure of non-reversible Finsler manifolds, this normed algebra had to be modified to include the cone of smooth semi-Lipschitz functions, resulting in a partial loss of the normed space and algebra structures. In order to achieve the desired result, we define new algebraic/quasi-metric structures to model the behavior of the aforementioned function space.