Isometries between the unit spheres of spaces of metrics

Abstract

Given a topological space $Z$, let $PM(Z)$ be the space of bounded continuous pseudometrics on $Z$, which is endowed with the sup-norm, and let $SPM(Z)$ be the unit sphere of $PM(Z)$. In this paper, we shall prove that for all non-degenerate compact metrizable spaces $X$ and $Y$, and for any surjective isometry $T : SPM(X) \to SPM(Y)$, there exists a homeomorphism $ϕ: Y \to X$ such that for any metric $d \in SPM(X)$ 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.