This is really more of a differential geometry question than a GR question. Is it possible to have torsion in two dimensions? I've seen statements implying that torsion is only of interest in 3 or more dimensions, but I don't see why. My understanding is that in two dimensions, you could not have a totally antisymmetric torsion tensor, and therefore it's not possible to have torsion in which parallel-transporting a tangent vector along a geodesic keeps it tangent to the geodesic. This doesn't seem the same as saying that torsion is impossible or of no interest in 2 dimensions.