Fixed-point set of finite group of diffeomorphisms is closed submanifold
Note that this will also show that the set of fixed points under a particular element of is a submanifold, because we can consider the cyclic subgroup generated by that element.
First step: Invariant Riemannian metric
The first step is to construct a Riemannian metric on the manifold that is invariant under the finite group. This is done as follows:
- Choose any Riemannian metric on
- Average this Riemannian metric under the action of
Second step: Nearby fixed points mean fixed geodesics
Fill this in later