To have a connection in GR one imposes several conditions. Among others, one condition is that the connection shall be torsion free, which leads to symmetric Christoffel symbols. My understanding is that this condition is imposed to maintain the theory as simple as possible. The absence of this condition would imply some physical phenomenon (spacetime with torsion, additionally to curvature), which is not observed (?). Further, one imposes metric compatibility, which leads to a vanishing covariant derivative of the metric. I assume that the reasons which make this property desirable are the same as above, but I cannot imagine any physical phenomenon related to the absence of this condition. So, my question is: what is the physical interpretation of a connection which is not compatible with the metric? I read in Sean Carrolls GR notes (p. 91), that to introduce fermions in GR one has to make use of a connection which is not metric compatible (called spin connection), why?