Slice theorem (differential geometry)

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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 extends to an invariant neighborhood of (viewed as a zero section) in 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 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[edit]

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[edit]


  1. ^ Audin 2004, Theorem I.2.1

External links[edit]