Slice theorem (differential geometry)
In differential geometry, the slice theorem states:[1] 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.
Idea of proof when G is compact
[edit]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.
See also
[edit]- Luna's slice theorem, an analogous result for reductive algebraic group actions on algebraic varieties
References
[edit]- ^ Audin 2004, Theorem I.2.1
External links
[edit]- On a proof of the existence of tubular neighborhoods
- Audin, Michèle (2004). Torus Actions on Symplectic Manifolds (in German). Birkhauser. doi:10.1007/978-3-0348-7960-6. ISBN 978-3-0348-7960-6. OCLC 863697782.