Lie groups, Affine Connections

1. Nov 6, 2009

zhangzujin

Let G be a Lie group. Show that there exists a unique affine connection such that $$\nabla X=0$$ for all left invariant vector fields. Show that this connection is torsion free iff the Lie algebra is Abelian.

2. Nov 6, 2009

Homework?

3. Nov 7, 2009

zhangzujin

Aha. Of course not. I'm just reading Riemannian Geometry by Petersen, interested in the exercises of that.

In fact, my major is PDEs.

4. Nov 7, 2009

zhentil

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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