Abstract
After an introduction which provides the reader with the basic notions and a number of examples, we show that a Lie group action gives rise to vector fields of a special type, called Killing vector fields. We prove the Orbit Theorem, which states that the distribution spanned by the Killing vector fields is integrable and that its integral manifolds coincide with the connected components of the orbits of the action. This way, every orbit gets endowed with the structure of an initial submanifold. After this general part, we limit our attention to the important special class of proper Lie group actions. Under this additional regularity assumption, one can prove the Tubular Neighbourhood Theorem (or Slice Theorem) which relates the action in a neighbourhood of an orbit to the isotropy representation at a point on that orbit and thus provides a normal form for the action near that orbit. As an important special case, we discuss free proper actions and related bundle structures. In the final two sections, we study invariant vector fields and make some elementary remarks on relative equilibria and relatively periodic integral curves. This is relevant for the study of Hamiltonian systems with symmetries in Chap. 10.
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Notes
- 1.
Or fundamental vector fields.
- 2.
Also known as the Slice Theorem.
- 3.
Another common name is isotropy group of m.
- 4.
In the case of a right action, Ψ m (ba)=Ψ m (a) iff b∈G m . Thus, \(\hat{\varPsi}_{m}\) is defined on the homogeneous space of right cosets by \(\hat{\varPsi}_{m}(G_{m} a) := \varPsi_{m}(a)\). It is equivariant with respect to the induced right translations \(\hat{\mathrm{R}}_{a}\).
- 5.
More precisely, a left (right) representation in case Ψ is a left (right) action.
- 6.
Subgroups H 1 and H 2 of G are said to be conjugate if H 2=aH 1 a −1 for some a∈G. To be conjugate is an equivalence relation on the set of subgroups of G. The equivalence classes are called conjugacy classes. The conjugacy class of a subgroup H in G will be denoted by [H].
- 7.
Given by matrix multiplication of SO(3)-matrices with elements of ℝ3.
- 8.
There is also a direct argument proving this, see Exercise 6.2.4.
- 9.
In particular, according to Proposition 3.5.21, they form the foliation associated with \(D^{\mathfrak{g}}\).
- 10.
Both of these statements follow as well by viewing O as the image of the submanifold \((G/G_{m},\hat{\varPsi}_{m})\) and using that \(\hat{\varPsi}_{m}\) is G-equivariant.
- 11.
Smoothness follows from the existence of local sections of the submersion (TM)↾O →NO.
- 12.
More generally, this definition applies when the domain is Hausdorff and the range is locally compact Hausdorff [53]. Compactness of subsets is understood with respect to the relative topology.
- 13.
The assumption that the submanifold be initial is made to ensure that the restricted action is smooth, cf. Example 6.1.2/7.
- 14.
That is, for all m 1,m 2∈M such that G⋅m 1≠G⋅m 2, there is f∈C ∞(M)G with f(m 1)≠f(m 2).
- 15.
In fact, the construction is designed so that the sequence of smooth functions, given by the partial sums, converges in an appropriate topology on C ∞(M). For the latter, see e.g. [211].
- 16.
For an alternative proof, see Exercise 6.3.1.
- 17.
The special case of a free compact Lie group action is due to Gleason [107].
- 18.
Any G m -invariant complement will do.
- 19.
More precisely, ψ m is an isomorphism of G-vector bundles; in particular, NO is globally trivial.
- 20.
A proper action of a discrete group is called a properly discontinuous action.
- 21.
Left (right) translations if Ψ is a left (right) action.
- 22.
In general, the converse does not hold. A local trivialization is more than a local section, because it is global along the fibre.
- 23.
The definition of locally trivial fibre bundle is obtained from that of vector bundle by replacing “vector space” by “manifold” and “linear mapping” by “smooth mapping”.
- 24.
That is, representing the conjugacy class of subgroups of G corresponding to σ.
- 25.
This follows also from the Transversal Mapping Theorem 1.8.2.
- 26.
E.g. in the theory of singular symplectic reduction.
- 27.
In the theory of semisimple Lie algebras, this cone is called a closed Weyl chamber.
- 28.
By an obvious modification of Definition 3.2.5, the notion of flow extends to the category of topological spaces and continuous mappings.
- 29.
Since the relative phases lie in N G (G γ ), this is both a left and a right coset.
- 30.
Which exists due to \((\varPsi_{a})'_{m} X_{m} = X_{\varPsi_{a}(m)} = X_{m}\) for all a∈G γ .
- 31.
These concepts are of topological nature and thus carry over word by word to topological flows.
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Rudolph, G., Schmidt, M. (2013). Lie Group Actions. In: Differential Geometry and Mathematical Physics. Theoretical and Mathematical Physics. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-5345-7_6
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