Smooth semi-Lipschitz functions and almost isometries of Finsler manifolds

TL;DR

This paper introduces a new approach using semi-Lipschitz functions to classify Finsler manifolds and characterize almost isometries, extending classical theorems to a geometric setting.

Contribution

It demonstrates that smooth semi-Lipschitz functions of constant less than one can classify Finsler manifolds and characterize their almost isometries.

Findings

Semi-Lipschitz functions form an order-algebraic structure capturing manifold features.

Functions of constant less than one classify Finsler manifolds.

Almost isometries are characterized via these functions, extending classical theorems.

Abstract

The convex cone $SC_{\mathrm{SLip}}^1(\mathcal{X})$ of real-valued smooth semi-Lipschitz functions on a Finsler manifold $\mathcal{X}$ is an order-algebraic structure that captures both the differentiable and the quasi-metric feature of $\mathcal{X}$. In this work we show that the subset of smooth semi-Lipschitz functions of constant strictly less than $1$, denoted $SC_{1^{-}}^1(\mathcal{X})$, can be used to classify Finsler manifolds and to characterize almost isometries between them, in the lines of the classical Banach-Stone and Mykers-Nakai theorems.