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.
Idea of proof when G is compact
Since G is compact, there exists an invariant metric; i.e., G acts as isometries. One then adopts the usual proof of the existence of a tubular neighborhood using this metric.
- Luna's slice theorem, an analogous result for reductive algebraic group actions on algebraic varieties
- Audin 2004, Theorem I.2.1
- On a proof of the existence of tubular neighborhoods
- Michele Audin, Torus actions on symplectic manifolds, Birkhauser, 2004
|This Differential geometry related article is a stub. You can help Wikipedia by expanding it.|