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

Statement

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

Note that this will also show that the set of fixed points under a particular element of $G$ 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 $M$
• Average this Riemannian metric under the action of $G$

Second step: Nearby fixed points mean fixed geodesics

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