Einstein-Cartan Theory vs Metric Affine Gravitation Theory Can anyone point out the real difference between Einstein-Cartan Theory and Metric Affine Gravitation Theory? Both of them rely on a pseudoriemannian metric $g$ and generalised affine connection $\Gamma$ (which is not the Christoffel symbol) and the introduction of a Torsion tensor $T$ but other than that it doesn't seem to distinguish the 2 theories. Is there something I'm missing? Any guidance appreciated!

I think it is the condition that in Einstein-Carman theory we demand

$\
abla_{k}^{\Gamma}g_{ij}=0$

Whereas in Metric Affine Gravitation Theory we define the Nonmetricity Tensor by

$\
abla_{k}^{\Gamma}g_{ij}=:C_{kij}\
eq0$