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A functional characterization of isometries between non-reversible Finsler manifolds

Francisco Venegas M

TL;DR

The paper addresses the problem of functionally characterizing isometries between non-reversible Finsler manifolds. It introduces an asymmetric function-space framework, constructing the extended asymmetric normed algebra $oxed{\mathcal{A}(\mathcal{X})}$ from the cone of smooth semi-Lipschitz functions, and proves a non-reversible Myers-Nakai-type theorem: an isomorphism of these algebras yields a Finsler isometry with explicit bi-semi-Lipschitz control. This establishes a Banach-Stone-type correspondence in fully asymmetric Finsler geometry and extends prior reversible/quasi-reversible results by leveraging quasi-metric and conic-algebraic structures. The work provides a robust functional-analytic mechanism to recover geometric isometries in non-symmetric settings, with potential implications for understanding the interplay between quasi-metric spaces and differential-geometric structures.

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.

A functional characterization of isometries between non-reversible Finsler manifolds

TL;DR

The paper addresses the problem of functionally characterizing isometries between non-reversible Finsler manifolds. It introduces an asymmetric function-space framework, constructing the extended asymmetric normed algebra from the cone of smooth semi-Lipschitz functions, and proves a non-reversible Myers-Nakai-type theorem: an isomorphism of these algebras yields a Finsler isometry with explicit bi-semi-Lipschitz control. This establishes a Banach-Stone-type correspondence in fully asymmetric Finsler geometry and extends prior reversible/quasi-reversible results by leveraging quasi-metric and conic-algebraic structures. The work provides a robust functional-analytic mechanism to recover geometric isometries in non-symmetric settings, with potential implications for understanding the interplay between quasi-metric spaces and differential-geometric structures.

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 -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.
Paper Structure (8 sections, 20 theorems, 39 equations)

This paper contains 8 sections, 20 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Finsler manifolds
  4. Quasi-metric spaces
  5. Semi-Lipschitz functions
  6. Algebraic challenges of non-reversibility
  7. New asymmetric structures
  8. Proof of the main result

Key Result

Theorem 1

Let $M$ and $N$ be two connected Riemannian manifolds. Then, $M$ and $N$ are isometric as Riemannian manifolds if and only if the spaces $C_b^1(M)$ and $C_b^1(N)$ are isometrically isomorphic as normed algebras. Moreover, every isometric isomorphism of normed algebras $T:C_b^1(N)\to C_b^1(M)$ must b

Theorems & Definitions (52)

  • Theorem 1: Myers-Nakai Theorem for Riemannian manifolds
  • Theorem 2: Myers-Nakai for quasi-reversible Finsler manifolds, isomorphic version
  • Theorem 3: Myers-Nakai for reversible Finsler manifolds, isometric version
  • Theorem 4: Desired result
  • Definition 5: Finsler manifold
  • Definition 6
  • Definition 7: Finsler isometry
  • Theorem 8
  • Definition 9: Quasi-metric space
  • Remark 10: Topology of a Finsler manifold
  • ...and 42 more