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Homework Help: Some questions on vector fields on Lie groups

  1. Oct 18, 2008 #1
    1. The problem statement, all variables and given/known data

    Let G be a Lie group with unit e. For every g in G let Lg: G -> G, h-> gh be left multiplication.

    a) Prove that the map G x TeG -> TG, (g,v) -> De Lg (v) is a diffeomorphism, and conclude G has a basis of vector fields.
    b) Prove for every v in TeG the is a unique vector field V on G that is left invariant (in the sense that the pullback of Lg on V is just V for all g in G) with Ve=v

    for the original problem statement, see

    https://www.math.uu.nl/people/looijeng/difftop06eng.pdf [Broken]

    Questions about this

    Lg (v) is just (gv), but I'm not sure how De transforms (gv). It is defined as the map that transforms a curve tangent to a point b to a curve tangent to e(b) on some other chart in G. If I understood this right, then De would send (gv) to the tangent vector at the origin in the direction of gv? - this last sentence of my own I did not fully understand.

    if someone could help clarify the problem a bit, not so much as give a solution... that would be nice.
     
    Last edited by a moderator: May 3, 2017
  2. jcsd
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