Question on Torsion free condition for Levi-Civita connection I was watching a video on Riemannian Geometry. The lecturer mentions that given the defining condition for a connection on a Riemannian manifold $M$ i.e. :
$$\nabla_X(Y) : \chi(M) \times \chi(M) \to \chi(M),$$ where $\chi(M)$ is the set of $C^{\infty}$ vector fields on M, the second of the Cartesian product from where Y comes as does the resulting quantity itself can be looked at as sections of a tangent bundle, while X is supposed to be looked at as a vector field. 
The metric compatibility condition as well as the linearity condition in X and the derivative rule in Y makes sense in such a case.
However, when we move onto the torsion-free condition
$$\nabla_{X} Y - \nabla_Y X = [X,Y]$$
inherent in Levi-Civita connections, both X,Y and the resulting $\nabla_{X} Y$ are all to be seen as vector fields. So the torsion free condition does not lend itself to the section of a bundle approach.
Can anyone please explain what this means? I was not able to fathom for myself.
Thanks. 

Having seen the part of the video you are referring to, Professor Morgan is talking about the more general context where you are defining a connection on a vector bundle $E$. That is a map 
\begin{align*}
\Gamma(TM)\times\Gamma(E) &\to \Gamma(E)\\
(X, s) &\mapsto \nabla_Xs.
\end{align*}
In this situation, the expressions on either side of the torsion-free condition aren't defined, they are only defined when $X$ and $Y$ are vector fields. That is, it only makes sense to talk about a torsion-free connection when your vector bundle is $E = TM$.
