# Fixed-point set of finite group of diffeomorphisms is closed submanifold

From Diffgeom

## Contents

## Statement

Let be a differential manifold and be a finite subgroup of the self-diffeomorphism group of . Then, the set of elements of that are fixed under *every* element of is a 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.

## Proof

### 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*