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 (viewed as a zero section) 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.
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.