Slice theorem (differential geometry) explained

M on which a Lie group G acts as diffeomorphisms, for any x in M , the map G/G _{x} \to M, [g] \mapsto g ⋅ x extends to an invariant neighborhood of G/G _{x} (viewed as a zero section) in G x _{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.

Idea of proof when G is compact

Since

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

See also

• Luna's slice theorem, an analogous result for reductive algebraic group actions on algebraic varieties

External links

• On a proof of the existence of tubular neighborhoods
• Book: Audin, Michèle . Torus Actions on Symplectic Manifolds . 2004 . 978-3-0348-7960-6 . 10.1007/978-3-0348-7960-6 . 863697782 . de . Birkhauser .