Slice theorem (differential geometry)In differential geometry, the slice theorem states:[1] given a manifold on which a Lie group acts as diffeomorphisms, for any in , 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 . The important application of the theorem is a proof of the fact that the quotient admits a manifold structure when 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 compactSince is compact, there exists an invariant metric; i.e., acts as isometries. One then adapts the usual proof of the existence of a tubular neighborhood using this metric. See also
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