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 struc
ture on $M$. We prove that $clq$ must be commutative as a $C^{ast}$ algebra i.e. $clqcong C(G)$ 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 $C(ISO(M))$.
Suppose that a compact quantum group Q acts faithfully and isomet- rically (in the sense of [10]) on a smooth compact, oriented, connected Riemannian manifold M . If the manifold is stably parallelizable then it is shown that the compact quantum grou
p is necessarily commutative as a C ast algebra i.e. Q = C(G) for some compact group G. Using this, it is also proved that the quantum isometry group of Rieffel deformation of such manifold M must be a Rieffel-Wang deformation of C(ISO(M))
