Slice theorem (differential geometry)

In differential geometry, the slice theorem states:[1] given a manifold

extends to an invariant neighborhood of

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

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.

is compact, there exists an invariant metric; i.e.,

One then adapts the usual proof of the existence of a tubular neighborhood using this metric.

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