Ivancevic_Applied-Diff-Geom

206 Applied Differential Geometry: A Modern Introductionneighborhood of e in G; if f : G → H is a smooth homomorphism of Liegroups, thenf ◦ exp G = exp H ◦T e f .Also, in this case (see [Chevalley (1955); Marsden and Ratiu (1999);Postnikov (1986)])exp(sξ) = γ ξ (s).Indeed, for fixed s ∈ R, the curve t ↦→ γ ξ (ts), which at t = 0 passes throughe, satisfies the differential equationddt γ ξ(ts) = sX ξ(γξ (ts) ) = X sξ(γξ (ts) ) .Since γ sξ (t) satisfies the same differential equation and passes through e att = 0, it follows that γ sξ (t) = γ ξ (st). Putting t = 1 induces exp(sξ) = γ ξ (s)[Marsden and Ratiu (1999)].Hence exp maps the line sξ in g onto the one–parameter subgroup γ ξ (s)of G, which is tangent to ξ at e. It follows from left invariance that theflow F ξ t of X satisfies F ξ t (g) = g exp(sξ).Globally, the exponential map exp, as given by (3.52), is a naturaloperation, i.e., for any morphism ϕ : G → H of Lie groups G and H and aLie functor F, the following diagram commutes [Postnikov (1986)]:F(G)exp❄GF(ϕ)ϕ✲ F(H)exp❄✲ HLet G 1 and G 2 be Lie groups with Lie algebras g 1 and g 2 . Then G 1 ×G 2is a Lie group with Lie algebra g 1 × g 2 , and the exponential map is givenby [Marsden and Ratiu (1999)].exp : g 1 × g 2 → G 1 × G 2 , (ξ 1 , ξ 2 ) ↦→ (exp 1 (ξ 1 ), exp 2 (ξ 2 )) .For example, in case of a nD vector space, or infinite–dimensional Banachspace, the exponential map is the identity.The unit circle in the complex plane S 1 = {z ∈ C : |z| = 1} is an AbelianLie group under multiplication. The tangent space T e S 1 is the imaginary