Let $M$ be a smooth manifold and let $G$ be a Lie group smoothly acting on $M$.
Then, under suitable assumptions (if $G$ acts freely and properly on $M$) we have a new smooth manifold $M/G$ corresponding to the orbits of the action.
I would like to know if there is a theorem that states (under suitable assumptions) that the set of fixed points $M^G$ can be equipped with a smooth manifold structure.
I suppose there is such a theorem, because $M^G$ is also the zero set of the infinitesimal generator of the action, which is a smooth vector field, so we have "smooth equations" describing it.
I will greatly appreciate any reference about this topic.
Thanks for your help!