Slice theorem (differential geometry)

In differential geometry, the slice theorem states:[1] given a manifold M on which a Lie group G acts as diffeomorphisms, for any x in M, the map ${\displaystyle G/G_{x}\to M,\,[g]\mapsto g\cdot x}$ extends to an invariant neighborhood of ${\displaystyle G/G_{x}}$ (viewed as a zero section) in ${\displaystyle G\times _{G_{x}}T_{x}M/T_{x}(G\cdot x)}$ 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 ${\displaystyle M/G}$ 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 adopts 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

References

1. Audin 2004, Theorem I.2.1 harvnb error: no target: CITEREFAudin2004 (help)

External links

• On a proof of the existence of tubular neighborhoods
• Michele Audin, Torus actions on symplectic manifolds, Birkhauser, 2004