Isometries of Products of Path-Connected Locally Uniquely Geodesic Metric Spaces with the Sup Metric are Reducible

TL;DR

This paper proves that isometries between products of path-connected, locally uniquely geodesic metric spaces with the sup metric are essentially reducible to reindexing and component-wise isometries, establishing a structural rigidity.

Contribution

It establishes that such isometries are necessarily composed of reindexing and component-wise isometries, showing a rigidity property of product spaces under the sup metric.

Findings

Isometries preserve the number of factors.

Isometries can be decomposed into reindexing and component isometries.

Structural rigidity of product spaces under the sup metric.

Abstract

Let $M_i$ and $N_i$ be path-connected locally uniquely geodesic metric spaces that are not points and $f:\prod_{i=1}^m M_i\to \prod_{i=1}^n N_i$ be an isometry where $\prod_{i=1}^n N_i$ and $\prod_{i=1}^m M_i$ are given the sup metric. Then $m=n$ and after reindexing $M_i$ is isometric to $N_i$ for all $i$. Moreover $f$ is a composition of an isometry that reindexes the factor spaces and an isometry that is a product of isometries $f_i:M_i\to N_i$.