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Diffgeom β

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

Contents

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

Fill this in later

Final step