Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 203at the end of the 19th Century, whose work is a striking synthesis of Lietheory, classical geometry, differential geometry and topology.These continuous groups, which originally appeared as symmetry groupsof differential equations, have over the years had a profound impact ondiverse areas such as algebraic topology, differential geometry, numericalanalysis, control theory, classical mechanics, quantum mechanics etc. Theyare now universally known as Lie groups.3.8.1 Definition of a Lie GroupA Lie group is a smooth (Banach) manifold M that has at the same time agroup G−structure consistent with its manifold M−structure in the sensethat group multiplicationand the group inversionµ : G × G → G, (g, h) ↦→ gh (3.50)ν : G → G, g ↦→ g −1 (3.51)are C k −maps [Chevalley (1955); Abraham et al. (1988); Marsden and Ratiu(1999); Puta (1993)]. A point e ∈ G is called the group identity element.For example, any nD Banach vector space V is an Abelian Lie groupwith group operations µ : V × V → V , µ(x, y) = x + y, and ν : V → V ,ν(x) = −x. The identity is just the zero vector. We call such a Lie groupa vector group.Let G and H be two Lie groups. A map G → H is said to be a morphismof Lie groups (or their smooth homomorphism) if it is their homomorphismas abstract groups and their smooth map as manifolds [Postnikov (1986)].All Lie groups and all their morphisms form the category LG (moreprecisely, there is a countable family of categories LG depending onC k −smoothness of the corresponding manifolds).Similarly, a group G which is at the same time a topological spaceis said to be a topological group if maps (3.50–3.51) are continuous, i.e.,C 0 −maps for it. The homomorphism G → H of topological groups is saidto be continuous if it is a continuous map. Topological groups and theircontinuous homomorphisms form the category T G.A topological group (as well as a smooth manifold) is not necessarilyHausdorff. A topological group G is Hausdorff iff its identity is closed. Asa corollary we have that every Lie group is a Hausdorff topological group(see [Postnikov (1986)]).