Characterization of a Banach-Finsler manifold in terms of the algebras of smooth functions

TL;DR

This paper establishes conditions under which the weak Finsler structure of certain complete Finsler manifolds can be uniquely determined by the algebra of bounded smooth functions, linking geometric and algebraic properties.

Contribution

It provides new criteria showing that the Finsler structure of various classes of complete Finsler manifolds is uniquely determined by their algebra of smooth functions.

Findings

Finsler structure of finite-dimensional complete manifolds determined by $C_b^k(M)$

Separable manifolds modeled on Banach spaces with bump functions have their structure determined by $C^k_b(M)$

WCG Banach spaces with smooth bump functions have their structure determined by $C^k_b(M)

Abstract

In this note we give sufficient conditions to ensure that the weak Finsler structure of a complete $C^k$ Finsler manifold $M$ is determined by the normed algebra $C_b^k(M)$ of all real-valued, bounded and $C^k$ smooth functions with bounded derivative defined on $M$. As a consequence, we obtain: (i) the Finsler structure of a finite-dimensional and complete $C^k$ Finsler manifold $M$ is determined by the algebra $C_b^k(M)$; (ii) the weak Finsler structure of a separable and complete $C^k$ Finsler manifold $M$ modeled on a Banach space with a Lipschitz and $C^k$ smooth bump function is determined by the algebra $C^k_b(M)$; (iii) the weak Finsler structure of a $C^k$ uniformly bumpable and complete $C^k$ Finsler manifold $M$ modeled on a Weakly Compactly Generated (WCG) Banach space with an (equivalent) $C^k$ smooth norm is determined by the algebra $C^k_b(M)$; and (iii) the isometric structure of a WCG Banach space $X$ with an $C^1$ smooth bump function is determined by the algebra $C_b^1(X)$.