Lie Group Actions
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
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Lie Group Actions
In Sect. 6.1, we define the notions of Lie group action and G-manifold and collect some of their elementary properties. There is a variety of derived notions fitting together to a geometric structure which will be studied in this chapter. In particular, a Lie group action gives rise to a special type of vector fields, so-called Killing vector fields,1 see Sect. 6.2. These vector fields span an integrable distribution whose integral manifolds coincide with the orbits of the group action. This way, every orbit is endowed with the structure of an initial submanifold. Starting from Sect. 6.3, we limit our attention to the important special class of proper group actions. Under this additional regularity assumption, one can prove the Tubular Neighbourhood Theorem2 which constitutes one of the basic tools of the theory of Lie group actions, see Sect. 6.4. It states that for every orbit there exists a G-invariant neighbourhood and a diffeomorphism identifying this neighbourhood G-equivariantly with a G-invariant neighbourhood of the zero section in the normal bundle of this orbit. In particular, we study the case of a free proper action in some detail, because it gives rise to interesting bundle structures. Next, in Sect. 6.6, we study elementary properties of the orbit space of a given Lie group action, and in Sect. 6.7 we discuss invariant vector fields. The latter notion is of basic importance for the study of physical systems with symmetries, see Chap. 10. In Sect. 6.8, we make some elementary remarks on relative equilibria and relatively periodic integral curves.
6.1 Basics Let M be a smooth manifold and let G be a Lie group. Let g be the Lie algebra of G. Starting from this chapter, elements of g will be denoted by A, B, . . . . Depending on the context, they will be viewed either as left-invariant vector fields on G or as elements of the tangent space T1 G. 1 Or
fundamental vector fields.
2 Also
known as the Slice Theorem.
G. Rudolph, M. Schmidt, Differential Geometry and Mathematical Physics, Theoretical and Mathematical Physics, DOI 10.1007/978-94-007-5345-7_6, © Springer Science+Business Media Dordrecht 2013
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Lie Group Actions
Definition 6.1.1 (Lie group action) 1. An action of G on M is a smooth mapping Ψ : G × M → M such that the induced mappings Ψa : M → M,
Ψa (m) := Ψ (a, m),
satisfy Ψ1 = idM and either Ψa ◦ Ψb = Ψab or Ψa ◦ Ψb = Ψba for all a, b ∈ G. In the first case, Ψ is called a left action and in the second case a right action. The triple (M, G, Ψ ) is referred to as a Lie group action and the pair (M, Ψ ) as a G-manifold. 2. A morphism of Lie group actions (M1 , G1 , Ψ 1 ) and (M2 , G2 , Ψ 2 ) (both left or both right) consists of a smooth mapping ϕ : M1 → M2 and a Lie group homomorphism : G1 → G2 such that ϕ ◦ Ψ 1 = Ψ 2 ◦ ( × ϕ).
(6.1.1)
One says that the mapping ϕ intertwines the actions and If G1 = G2 and is the identical mapping, ϕ is also called equivariant or, equivalently, a morphism of the G-manifolds (M1 , Ψ 1 ) and (M2 , Ψ 2 ). Ψ1
Ψ 2.
Since Ψ
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