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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 $G/G_x \to M, \, \left(\right) \mapsto g \cdot x$ extends to an invariant neighborhood of $G/G_x$ (viewed as a zero section) in $G \times_ T_x M / T_x\left(G \cdot x\right)$ 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 adopts the usual proof of the existence of a tubular neighborhood using this metric.

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