A note on geometric characterization of quantum isometries of classical manifolds

TL;DR

This paper explores the relationship between quantum group actions and classical Riemannian manifolds, providing conditions under which quantum isometries are constrained or non-existent, thus deepening understanding of quantum symmetries in geometry.

Contribution

It offers a partial converse to known results, showing that orientation-preserving actions imply certain preservation properties, and provides an alternative proof that no quantum isometry exists for connected manifolds.

Findings

Quantum group actions preserving Laplacian imply preservation of Riemannian inner product.

Orientation-preserving actions lead to specific geometric constraints.

No quantum isometry exists for compact, connected Riemannian manifolds.

Abstract

If a compact quantum group acts isometrically on a (possibly discon- nected) compact smooth Riemannian manifold such that the action commutes with the Laplacian then it is known that the differential of the action preserves Rieman- nian inner product on forms. In this note, we prove a partial converse to this, under the additional assumption that the manifold is ori- ented and the action preserves the orientation in a suitable sense. Using this an alternative line of arguments is given for proving that there is no quantum isometry for a compact, connected, Riemannian manifold.