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Lie groups, Affine Connections

  1. Nov 6, 2009 #1
    Let G be a Lie group. Show that there exists a unique affine connection such that [tex]\nabla X=0[/tex] for all left invariant vector fields. Show that this connection is torsion free iff the Lie algebra is Abelian.
  2. jcsd
  3. Nov 6, 2009 #2
  4. Nov 7, 2009 #3
    Aha. Of course not. I'm just reading Riemannian Geometry by Petersen, interested in the exercises of that.

    In fact, my major is PDEs.
  5. Nov 7, 2009 #4
    The second statement shouldn't be bad, if you define the torsion tensor in terms of the connection and the commutator (i.e. show that [X,Y] is identically zero if and only if the Lie algebra is Abelian - shouldn't be too hard :) ).

    For the first part, why not define the connection to be zero at the identity, and then drag all your vectors back there by left translation?
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