2
$\begingroup$

Let a finite group $G$ act smoothly on a smooth manifold $M$ with one fixed point $p$. Then, by the Slice theorem, we are able to find a $G$-equivariant neighborhood, $U_p$, of $p$ which is $G$-diffeomorphic to $T_pM$ (it is endowed with an $\mathbb{R}G$-module structure arising from group action and we can tell about $G$ acting on it). Is it true that $T_pM$ is a faithful $\mathbb{R}G$-module? Ore, to put it differently since we have the $G$-diffeomorphism from $U_p$ to $T_pM$, is the action of $G$ on $U_p$ faithful?

$\endgroup$
1
$\begingroup$

Yes, this follows from the existence of a $G$-invariant Riemannian metric on $M$: if an isometry $\phi$ of a connected Riemannian manifold acts trivially on some $T_pM$ then $\phi=id$. (You prove this by looking at the exponential map.)

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.