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 G / G x → M , [ g ] ↦ g ⋅ x extends to an invariant neighborhood of G / G x (viewed as a zero section) in G × G x T x M / T x ( G ⋅ x ) 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 M / G 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.