Lie Group Methods *

Definition 1.1. [Ol] An m-dimensional manifold M is a topological space covered by a collection of open subsets Wα ⊂M (coordinate charts) and maps Xα : Wα → Vα ⊂ R one-to-one and onto, where Vα is an open, connected subset of R. (Wα,Xα) define coordinates on M. M is a smooth manifold if the maps Xαβ = Xβ ◦X−1 α , are smooth where they are defined, i.e. on Xα(Wα ∩Wβ) to Xβ(Wα ∩Wβ). Example 1.2. R is a m-dimensional manifold covered with a single chart. Example 1.3. The unit sphere Sm−1 := {x ∈ R | ∑m i=1 x 2 i = 1} is a m−1dimensional manifold covered with two charts obtained by omitting the north and south poles respectively. The coordinate maps are obtained considering the stereographic projection from the north and south pole respectively.