What's the difference between Torsion and Curvature in Einstein-Cartan theory? I have a very simple question:
In the phrase "gravity is the manifestation of curvature", I want to know its analogue in Einstein Cartan theory where torsion is non-vanishing.
So basically I want to fill the gap in the following phrase: $\dots$ is the manifestation of Torsion.
And could someone, please, explain to me why Torsion has the same statue as: 1) Curvature when we consider the field equations of E.C.T. $$ R_{ab}-\frac{1}{2} R = \kappa P_{a b} $$ $$T_{a b}{ }^{c}+g_{a}{ }^{c} T_{b d}{ }^{d}-g_{b}{ }^{c} T_{a d}{ }^{d}=\kappa \sigma_{a b}{ }^{c} $$
where $P_{ab}$ is the energy momentum density, $ \sigma_{a b}{ }^{c}$ is what we call a spin tensor and $g_{a}{ }^{c}$ is the metric of a Riemann-Cartan space.
2) And a connection when we consider the affine connection $$ \Gamma= \tilde{\Gamma}+ K(T) $$
Where $\tilde{\Gamma}$ is the Levi-Civita connection and $K$ is the contorsion.
Am asking this question because there is no such thing in GR.
N.B.: I know the geometrical meaning of Torsion and that the so-called spin current is related to it.
 A: I would be very careful with the phrase "gravity is the manifestation of curvature". Because if one falls towards a massive body --- this observation is typically associated (by laymans)  with gravity ---  it is actually not a symptome of curvature, it simply might mean that one is in a "wrong" coordinate/reference system that provides one this impression.
However, if one includes in the concept of gravity the occurrence of tidal forces, then one could indeed attribute that to curvature. In this sense one could say: "gravity is the manifestation of curvature".
Actually I would prefer to say:
Curvature is generated by momentum & energy (density)   whereas
Torsion   is generated by spin (density).
The effect of torsion is actually very weak. As the 2. equation (also 2. equation of your post) which links torsion with spin is an algebraic equation torsion cannot propagate. One would find torsion only at places with significant spin density, for instance inside exotic astronomical objects like a neutron star or in the first instants after the big-bang.
Under the assumption that the torsion tensor is totally antisymmetric, autoparallelism equation and geodesic equation are the same. Therefore the torsion would not influence the motion of a particle in a place of high spin density.
The only effect torsion produces is spin precession. So one could tentatively say: Spin precession is a manifestation of torsion.
But as there are other effects that can produce spin precession --- and mostly probably stronger than the one generated by the torsion --- one should be extremely careful in the expression of such an assertion. Because the effect (spin precession) and the concept (torsion) are not uniquely related.
The most pursued approach in handling with torsion is to try to eliminate it from the equations, because torsion makes the math quite complicated. In this respect there would be no manifestation at all.
