My question is to the same effect as the question Difference between torsion and Lie bracket
Let's suppose we have a situation where the lie bracket is nonzero but the torsion tensor vanishes.
Let $[X,Y]=A≠0$
Since the torsion tensor vanishes and $T(X,Y):=\nabla_XY-\nabla_YX-[X,Y]$ we have $A=\nabla_XY-\nabla_YX$
In a situation like this, does $\nabla_XY-\nabla_YX$ "compensate" for the non-closing of the flow lines?
Whenever I've identified a connection to be torsion free I always end up with $0=\nabla_XY-\nabla_YX$. I know this can't be the only type of situation where a connection is torsion-free, and thus I want to better understand the situations where $0≠\nabla_XY-\nabla_YX$