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dg2test2 - Differential Geometry 2 Test 2 Define what is...

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Unformatted text preview: Differential Geometry 2 Test 2 Define what is meant by a Lie group G and by its Lie algebra g ; define also the exponential map exp : g → G . (a) Show that R → G : u 7→ exp( uξ ) is the integral curve of ξ ∈ g through the identity e ∈ G . (b) Prove that there exist open sets (0 3 ) U ⊂ g and ( e 3 ) V ⊂ G such that exp is a diffeomorphism from U to V . Note : Smoothness of the exponential map may be assumed. Solutions : A Lie group G is both a group and a smooth manifold, these structures being compatible in the sense that the group operation G × G → G is a smooth map (inversion G → G being also smooth as a consequence). Its Lie algebra g comprises all vector fields on G that are left-invariant; it is a Lie algebra under the usual bracket of vector fields. If ξ ∈ g is a left- invariant vector field on G then exp( ξ ) ∈ G is defined by exp( ξ ) = γ ξ (1) where γ ξ : R → G is the (necessarily complete) integral curve of ξ through the identity e ....
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