Slice theorem (differential geometry)

In differential geometry, the slice theorem states:^[1] given a manifold ${\displaystyle M}$ on which a Lie group ${\displaystyle G}$ acts as diffeomorphisms, for any ${\displaystyle x}$ in ${\displaystyle M}$, 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 ${\displaystyle x}$.

The important application of the theorem is a proof of the fact that the quotient ${\displaystyle M/G}$ admits a manifold structure when ${\displaystyle G}$ 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 ${\displaystyle G}$ is compact, there exists an invariant metric; i.e., ${\displaystyle G}$ 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