In the presence of torsion, is it correct to say that the metric doesn't give rise to a unique connection? So, if we were using, say ECKS theory, which unlike GR includes torsion, in order to find the connection, we'd need not only the metric, but a specification of the torsion. My thinking is that the metric gives the symmetric part of the Christoffel symbols of the second kind, but in the presence of torsion, there is also an asymmetric part for these symbols which is given by the torsion. However, I could use a sanity check here. A related question: if we define a geodesic by saying that it parallel transports its own tangent vector, is the geodesic equation unaffected by the presence of the torsion? So the metric still gives us the geodesic equation, the unspecified torsion terms don't matter?