Slice theorem (differential geometry)

Short description: On extending a Lie group action on a manifold to an equivariant diffeomorphism

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 $G/G_x\to M,[g]\mapsto g\cdot x$ extends to an invariant neighborhood of $G/G_x$ (viewed as a zero section) in $G{\times }_{G_x}{T}_{x}M/T_x(G\cdot x)$ 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
• Audin, Michèle (2004) (in de). Torus Actions on Symplectic Manifolds. Birkhauser. doi:10.1007/978-3-0348-7960-6. ISBN 978-3-0348-7960-6. OCLC 863697782.