Is There a Unique Torsion-Free Affine Connection on a Lie Group?

[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.

 

[hamster143]

Homework?

 

[zhangzujin]

Homework?

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

In fact, my major is PDEs.

 

[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?