Abstract
In this chapter, we present a concise introduction to the theory of proper and isometric actions. We begin with basic definitions and a quick primer on fiber bundles, leading to the Slice Theorem 3.49 and the Tubular Neighborhood Theorem 3.57. These fundamental results are used throughout the rest of the book; in particular, to establish a strong correspondence between proper and isometric actions in Sect. 3.3. Finally, the stratification of a manifold by orbit types of a proper action is studied in Sect. 3.5.
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Notes
- 1.
Analogous maps can be defined for a right G-action, simply replacing μ(g, x) by μ(x, g).
- 2.
Notice that μ G is isomorphic to G only if the action μ is effective.
- 3.
For more details on this action, see Example 6.48.
- 4.
A posteriori, this condition means that the submanifold S is transverse to the orbit G(x).
- 5.
In fact, since (3.5) is a proper map between locally compact Hausdorff spaces, it is also closed, i.e., maps closed subsets to closed subsets. Thus, its image \(\mathcal{R} =\{ (x,y) \in M \times M: G(x) = G(y)\}\) is closed. As \(\rho: M \rightarrow M/G\) is an open map, \(\rho (M \times M\setminus \mathcal{R})\) is open. This is easily seen to be the complement of the diagonal in \(M/G \times M/G\), which is hence closed, proving that M∕G is Hausdorff.
- 6.
More generally, two (either left or right) actions μ 1 and μ 2 on M are said to commute if the induced transformations (3.1) on M satisfy \((\mu _{1})^{g_{1}} \circ (\mu _{2})^{g_{2}} = (\mu _{2})^{g_{2}} \circ (\mu _{1})^{g_{1}}\) for all g i ∈ G i .
- 7.
- 8.
This is a consequence of the so-called Kronecker Approximation Theorem.
- 9.
- 10.
Recall that \(\overline{\mathbb{C}P}^{2}\) denotes the complex projective plane \(\mathbb{C}P^{2}\) endowed with the orientation opposite to the standard. It is well-known that the connected sum \(\mathbb{C}P^{2}\#\overline{\mathbb{C}P}^{2}\) is the only nontrivial S 2-bundle over S 2, since such bundles are classified by \(\pi _{1}(\mathrm{Diff}(S^{2}))\mathop{\cong}\pi _{1}(\mathrm{SO}(3))\mathop{\cong}\mathbb{Z}_{2}\).
- 11.
This theorem gives a criterion for convergence of continuous maps in the compact-open topology. More precisely, a sequence of continuous functions \(\{g_{n}: K \rightarrow B\}\) between compact metric spaces K and B admits a uniformly convergent subsequence if {g n } is equicontinuous, i.e., for every \(\varepsilon > 0\) there exists δ > 0 such that \(\mathrm{dist}(g_{n}(x),g_{n}(y)) <\varepsilon\) for all dist(x, y) < δ, x, y ∈ K, \(n \in \mathbb{N}\).
- 12.
This means that any open cover of M∕G admits a locally finite refinement.
- 13.
Note that completeness can be guaranteed if, e.g., M is compact.
- 14.
Note that the assumption that γ realizes the distance between the orbits G(γ(0)) and G(γ(1)) is stronger than γ being a minimal geodesic segment between its endpoints γ(0) and γ(1).
- 15.
A metric space is a length metric space if the distance between any two points is realized by a shortest curve, called a geodesic.
- 16.
- 17.
More generally, if μ is not free, then \(\pi: M \rightarrow M/G\) is a submetry, which is a generalization of submersions to metric spaces.
- 18.
It is important to notice that singular orbits are smooth submanifolds, whose singular nature is simply to have a lower dimension.
- 19.
To our knowledge, the notion of local orbit types was introduced in Duistermaat and Kolk [79].
- 20.
If the action is isometric, this simply means that the slice representations of G x and G y are equivalent, i.e., related by an equivariant linear map \(\phi: \nu _{x}G(x) \rightarrow \nu _{y}G(y)\).
- 21.
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Alexandrino, M.M., Bettiol, R.G. (2015). Proper and Isometric Actions. In: Lie Groups and Geometric Aspects of Isometric Actions. Springer, Cham. https://doi.org/10.1007/978-3-319-16613-1_3
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