Ivancevic_Applied-Diff-Geom

208 Applied Differential Geometry: A Modern Introductionand g, h ∈ G, (i) φ(e, x) = x and (ii) φ (g, φ(h, x)) = φ(gh, x). In otherwords, letting φ g : x ∈ M ↦→ φ g (x) = φ(g, x) ∈ M, we have (i’) φ e = id Mand (ii’) φ g ◦ φ h = φ gh . φ g is a diffeomorphism, since (φ g ) −1 = φ g −1. Wesay that the map g ∈ G ↦→ φ g ∈ Diff(M) is a homomorphism of G into thegroup of diffeomorphisms of M. In case that M is a vector space and eachφ g is a linear operator, the function of G on M is called a representationof G on M [Puta (1993)]An action φ of G on M is said to be transitive group action, if for everyx, y ∈ M, there is g ∈ G such that φ(g, x) = y; effective group action, ifφ g = id M implies g = e, that is g ↦→ φ g is 1–1; and free group action, if foreach x ∈ M, g ↦→ φ g (x) is 1–1.For example,(1) G = R acts on M = R by translations; explicitly,φ : G × M → M, φ(s, x) = x + s.Then for x ∈ R, O x = R. Hence M/G is a single point, and the actionis transitive and free.(2) A complete flow φ t of a vector–field X on M gives an action of R onM, namely(t, x) ∈ R × M ↦→ φ t (x) ∈ M.(3) Left translation L g : G → G defines an effective action of G on itself.It is also transitive.(4) The coadjoint action of G on g ∗ is given byAd ∗ : (g, α) ∈ G × g ∗ ↦→ Ad ∗ g −1(α) = ( T e (R g −1 ◦ L g ) ) ∗α ∈ g ∗ .Let φ be an action of G on M. For x ∈ M the orbit of x is defined byO x = {φ g (x)|g ∈ G} ⊂ Mand the isotropy group of φ at x is given byG x = {g ∈ G|φ(g, x) = x} ⊂ G.An action φ of G on a manifold M defines an equivalence relation onM by the relation belonging to the same orbit; explicitly, for x, y ∈ M, wewrite x ∼ y if there exists a g ∈ G such that φ(g, x) = y, that is, if y ∈ O x .The set of all orbits M/G is called the group orbit space.