Einstein-Cartan Theory vs Metric Affine Gravitation Theory

Question

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!

Answer 1

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

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

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

$\nabla_{k}^{\Gamma}g_{ij}=:C_{kij}\neq0$