# Existence and rigidity of quantum isometry groups for compact metric spaces

## TL;DR

This paper establishes the existence of quantum isometry groups for certain classes of compact metric spaces, including geodesic metrics on Riemannian manifolds and spaces with uniform measures, and shows their classical nature in some cases.

## Contribution

It introduces the existence of quantum isometry groups for new classes of metric spaces and demonstrates their classical properties in specific instances.

## Key findings

- Quantum isometry groups exist for geodesic metrics on compact Riemannian manifolds.
- Quantum isometry groups exist for metric spaces with a uniform probability measure.
- In some cases, quantum isometry groups are classical, coinciding with traditional isometry groups.

## Abstract

We prove the existence of a quantum isometry groups for new classes of metric spaces: (i) geodesic metrics for compact connected Riemannian manifolds (possibly with boundary) and (ii) metric spaces admitting a uniformly distributed probability measure. In the former case it also follows from recent results of the second author that the quantum isometry group is classical, i.e. the commutative $C^*$-algebra of continuous functions on the Riemannian isometry group.