Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 207axis, and we identify R with T e S 1 by t ↦→ 2πit. With this identification,the exponential map exp : R → S 1 is given by exp(t) = e 2πit .The nD torus T n = S 1 ×···×S 1 (n times) is an Abelian Lie group. Theexponential map exp : R n → T n is given bySince S 1 = R/Z, it follows thatexp(t 1 , ..., t n ) = (e 2πit1 , ..., e 2πitn ).T n = R n /Z n ,the projection R n → T n being given by the exp map (see [Marsden andRatiu (1999); Postnikov (1986)]).For every g ∈ G, the mapAd g = T e(Rg −1 ◦ L g): g → gis called the adjoint map (or operator) associated with g.For each ξ ∈ g and g ∈ G we haveexp (Ad g ξ) = g (exp ξ) g −1 .The relation between the adjoint map and the Lie bracket is the following:For all ξ, η ∈ g we haveddt∣ Ad exp(tξ) η = [ξ, η].t=0A Lie subgroup H of G is a subgroup H of G which is also a submanifoldof G. Then h is a Lie subalgebra of g and moreover h = {ξ ∈ g| exp(tξ) ∈ H,for all t ∈ R}.Recall that one can characterize Lebesgue measure up to a multiplicativeconstant on R n by its invariance under translations. Similarly, on a locallycompact group there is a unique (up to a nonzero multiplicative constant)left–invariant measure, called Haar measure. For Lie groups the existenceof such measures is especially simple [Marsden and Ratiu (1999)]: Let Gbe a Lie group. Then there is a volume form Ub5, unique up to nonzeromultiplicative constants, that is left–invariant. If G is compact, Ub5 is rightinvariant as well.3.8.2 Actions of Lie Groups on Smooth ManifoldsLet M be a smooth manifold. An action of a Lie group G (with the unitelement e) on M is a smooth map φ : G × M → M, such that for all x ∈ M