Isometries between the unit spheres of spaces of metrics

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 $\phi :Y\to X$ such that for any metric $d\in SPM(X)$ and for any pair of points $(x,y)\in Y^2$, $T(d)(x,y)=d(\phi (x),\phi (y))$. As a corollary, we can solve a variant of Tingley’s problem on spaces of metics.