Link between affine connections and Lie brackets

Question

I am studying the geometric theory of differentiable manifolds and I came across a sentence which I didn't fully understand.

I am referring to the wikipedia page about Affine connections. Where there is written:

The main invariants of an affine connection are its torsion and its curvature. The torsion measures how closely the Lie bracket of vector fields can be recovered from the affine connection.

I didn't really get what is the link between an affine connection and the Lie Bracket.

Could you help me with that?

Thanks a lot.

Answer 1

If $\nabla$ is an affine connection, its torsion tensor is

$$T(X, Y) = \nabla_X Y - \nabla_Y X - [X, Y]$$

Hence we see that $[X, Y] = \nabla_X Y - \nabla_Y X - T(X, Y)$.

(Note that if $\nabla$ is the Levi-Civita connection of a Riemannian metric, then $T = 0$).