Metric-Affine Manifold
  We call a manifold with torsion and nonmetricity the metric-affine manifold. The nonmetricity leads to a difference between the auto parallel line and the extreme line, and to a change in the expression of the Frenet transport and moving basis. The torsion leads to a change in the Killing equation. We also need to add a similar equation for the connection.   The analysis of the Frenet transport leads to the concept of the Cartan transport and an introduction of the connection compatible with the metric tensor. The dynamics of a particle follows to the Cartan transport. We need additional physical constraints to make a nonmetricity observable. 
Aleks Kleyn