Isometries between the unit spheres of spaces of metrics

Katsuhisa Koshino

paper

Isometries between the unit spheres of spaces of metrics

arXiv:2606.25558

Abstract

Given a topological space $Z$ , let $P M (Z)$ be the space of bounded continuous pseudometrics on $Z$ , which is endowed with the sup-norm, and let $S P M (Z)$ be the unit sphere of $P M (Z)$ . In this paper, we shall prove that for all non-degenerate compact metrizable spaces $X$ and $Y$ , and for any surjective isometry $T : S P M (X) \to S P M (Y)$ , there exists a homeomorphism $ϕ : Y \to X$ such that for any metric $d \in S P M (X)$ and for any pair of points $(x, y) \in Y^{2}$ , $T (d) (x, y) = d (ϕ (x), ϕ (y))$ . As a corollary, we can solve a variant of Tingley's problem on spaces of metics.