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 $$G/G_x \to M, \, [g] \mapsto g \cdot x$$ 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.