Torsion of particular form

Question

Let $M$ be a 2D smooth manifold and let $\nabla$ be a flat connection on $M$. Denote by $T$ its torsion tensor. In a certain application, torsions of the form $$ T(X,Y) = X(f)\, Y - Y(f)\, X, $$
where $f$ is a real-valued function on $M$, seem to play an important role. I wondered whether a torsion of that particular form has a name, and whether it is known to be relevant in other contexts.