About: Slice theorem (differential geometry)

In differential geometry, the slice theorem states: given a manifold M on which a Lie group G acts as diffeomorphisms, for any x in M, 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 x. The important application of the theorem is a proof of the fact that the quotient admits a manifold structure when G 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.