Quantum rigidity of negatively curved manifolds

TL;DR

The paper proves that any quantum symmetry acting on a negatively curved manifold's metric space must be classical, aligning with the isometry group, thus limiting quantum symmetries in this geometric setting.

Contribution

It demonstrates that negatively curved manifolds do not admit non-classical quantum symmetries, answering a question about quantum isometry groups in such geometries.

Findings

Quantum actions on negatively curved manifolds are classical.

Quantum symmetries are constrained by negative curvature.

The result partially answers Goswami's question.

Abstract

We show that an isometric action of a compact quantum group on the underlying geodesic metric space of a compact connected Riemannian manifold $(M,g)$ with strictly negative curvature is automatically classical, in the sense that it factors through the action of the isometry group of $(M,g)$. This partially answers a question by D. Goswami.