Non-existence of faithful isometric action of compact quantum groups on compact, connected Riemannian manifolds

Abstract

Suppose that a compact quantum group $\clq$ acts faithfully on a smooth, compact, connected manifold $M$, i.e. has a $C^*$ (co)-action $\alpha$ on $C(M)$, such that the action $\alpha$ is isometric in the sense of \cite{Goswami} for some Riemannian structure on $M$. We prove that $\clq$ must be commutative as a $C^{\ast}$ algebra i.e. $\clq\cong C(G)$ for some compact group $G$ acting smoothly on $M$. In particular, the quantum isometry group of $M$ (in the sense of \cite{Goswami}) coincides with $C(ISO(M))$.