Slice theorem (differential geometry)

In differential geometry, the slice theorem states: given a manifold on which a Lie group ' acts as diffeomorphisms, for any ' in ', the map extends to an invariant neighborhood of in so that it defines an equivariant diffeomorphism from the neighborhood to its image, which contains the orbit of '.
The important application of the theorem is a proof of the fact that the quotient admits a manifold structure when is compact and the action is free.
In algebraic geometry, there is an analog of the slice theorem; it is called Luna's slice theorem.

Idea of proof when ''G'' is compact

Since ' is compact, there exists an invariant metric; i.e., ' acts as isometries. One then adapts the usual proof of the existence of a tubular neighborhood using this metric.