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dg2test3 - Dierential Geometry 2 Test 3 Dene what is meant...

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Differential Geometry 2 Test 3 Define what is meant by a Lie group homomorphism φ : G H and con- struct its derivative φ # : g h . (a) Show that φ exp G = exp H φ # and explain briefly why φ is uniquely determined by its derivative φ # when G is connected. (b) Show that if ϕ : g h is a Lie algebra homomorphism then there need not exist φ : G H such that φ # = ϕ . Suggestion : (b) Let G = H = S 1 . Let ϕ : Lie( S 1 ) Lie( S 1 ) be given by ϕ = 1 2 Id; assume ϕ = φ # and take the square. Solutions : When G and H are Lie groups, a Lie group homomorphism is a smooth, group homomorphism from G to H . Being a group homomor- phism, φ maps e = e G to e = e H ; being smooth, φ has a tangent map φ * : T e G T e H . Now the derivative φ # : g h is defined to be the com- posite g T e G φ * T e H h in which the first map is evaluation at e G and the last is the inverse of evaluation at e H . (a) Let ξ g have integral curve γ through e G in G : if t R then it follows that ( φ γ ) ( t ) = φ * ( γ ( t )) = φ * ( ξ γ ( t ) ) = φ * ( L γ (
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