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arXiv · 2606.25558

Isometries between the unit spheres of spaces of metrics

Katsuhisa Koshino
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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 $\phi : Y \to X$ 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.

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Katsuhisa Koshino. 2026-06-24. Isometries between the unit spheres of spaces of metrics. https://arxiv.org/abs/2606.25558

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