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arXiv · 2609.22201

Modifications of the Continuous Gromov--Hausdorff Distance

A. A. Vikhrov
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Abstract

This work studies modifications of the Gromov--Hausdorff distance in which the infimum is taken over pairs of morphisms of a given class $\FF$ --- the class of morphisms of a subcategory of the category of metric spaces (the $\FF$-Gromov--Hausdorff distance). We prove that the $\FF$-distance depends Lipschitz-continuously on the class $\FF$ in the Hausdorff metric on morphisms. As a consequence, for second countable $C^k$-manifolds equipped with a metric compatible with the topology, the smooth distance coincides with the continuous one: $\dGH{C_k} = \dGH{C}$. We also introduce the partially continuous and partially locally constant distances (continuity is weakened to continuity on an open subset of the support of full measure) and prove that on the class of metric spaces they coincide with the classical Gromov--Hausdorff distance.

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A. A. Vikhrov. 2026-08-31. Modifications of the Continuous Gromov--Hausdorff Distance. https://arxiv.org/abs/2609.22201

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