Fill in the response. Read the following conversation:
+ What is meant by a torsion-free spin connection?.
+ This definition should be taken as defining the torsion-free spin connection, since, by convention, the Christoffel symbols are derived from the Levi-Civita connection, which is the unique metric compatible, torsion-free connection on a Riemannian Manifold..
+ Is there anything else interesting about the torsion-free spin connection?.
+ In general, there is no restriction: the spin connection may also contain torsion..
+ Where is the torsion located?.
+ Note that formula_15 using the gravitational covariant derivative formula_16 of the contravariant vector formula_17..
+ What other aspects of the connection are notable?.
+ The spin connection may be written purely in terms of the vierbein field as  which by definition is anti-symmetric in its internal indices formula_19..
+ Is there anything else important about the torsion-free spin connection?.
+ 
The spin connection formula_20 defines a covariant derivative formula_21 on generalized tensors.