Rigidity of action of compact quantum groups III: the general case

Abstract: If a compact quantum group acts faithfully and smoothly (in the sense of Goswami 2009) on a smooth, compact, oriented, connected Riemannian manifold such that the action induces a natural bimodule morphism on the module of sections of the co-tangent bundle, then it is proved that the quantum group is necessarily commutative as a $C^{*}$ algebra i.e. isomorphic with $ C(G)$ for some compact group $G$. From this, we deduce that the quantum isometry group of such a manifold M coincides with $C(ISO(M))$ where $ISO(M) $ is the group of (classical) isometries, i.e. there is no genuine quantum isometry of such a manifold.