Metric non symmetric connection in the tangent bundle?

Question

Let $\Sigma$ be a surface endowed with a Riemannian metric $g.$ According to the fundamental theorem of Riemannian geometry, there exist a unique $\nabla$ symmetric connection ( i.e. torsionless) in the tangent bundle of $\Sigma$ that preserves the metric, that is $$ \nabla g \equiv 0.$$ Are there any examples of non symmetric connections that preserve a metrig $g$ on a surface $\Sigma?$ What if $g$ is flat?

Answer 1

A slight extension of what you refer to as the "fundamental theorem of Riemannian geometry" says that mappsing linear connections on $TM$ which are compatible with a Riemannian metric $g$ to their torsion is an isomorphism between metric connections and smooth sections of the bundle $\Lambda^2T^*M\otimes TM$. So for any skew symmetric tensor field $T$, there is a unique metric connection with torsion $T$. It is not difficult to write out an explicit formula for the contorsion needed to modify the Levi-Civita connection to a metric connection with torsion $T$.