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

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 $f_i:M_i\to N_i$.