# Local version of a slice (for a Lie group action)

Let $\Upsilon: G \times M \to M$ be a smooth action of a Lie group $G$ on a manifold $M$.

Isenberg and Marsden (1982) define a slice at $m \in M$ as a submanifold $S \subseteq M$ containing $m$ such that

• $S$ is invariant under the induced action of $G_m$.
• $(\Upsilon_g S) \cap S \neq \emptyset$ implies $g \in G_m$.
• There exists a local section $\chi: G/G_m \supseteq U \to G$ defined in a neighborhood $U$ of the identity coset such that the map $$\chi^S: U \times S \to M, \qquad ([g], s) \mapsto \chi([g]) \cdot s$$ is a diffeomorphism onto a neighborhood $V \subseteq M$ of $m$.

The same definition can be found in KrieglMichor1997, with a short comment that the usual finite-dimensional definition is too rigid for infinite-dimensional applications. What are concrete examples where this weaker definition of a slice is required and the usual finite-dimensional version does not apply?

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I have also found this notion of a slice also in the work of Ebin and, moreover, in connection with the orbit space of gauge theory. So it seams to be the 'standard' definition. But to my knowledge, in all of these cases the slice theorem is proofed in the usual way by a tubular neighborhood - which then would lead to the finite-dimensional notion of a slice?! –  Tobias Diez Nov 7 '13 at 22:22
I think the original (finite-dimensional, compact-Lie-group version) of the slice theorem was proved by Mostow in this paper: jstor.org/stable/1970055 –  alvarezpaiva Nov 8 '13 at 8:58