# Geodesics and actions in affine gravity

It is a well known result in Riemannian geometry that geodesics are also the curves that minimize length (or extremize it anyway, I know what you're gonna say @0celo7), with a very similar result in Lorentzian geometry, where for instance the Nambu-Goto action

$$S = -m \int_{t_0}^{t_1} ds = -m \int_{t'_0}^{t'_1} \sqrt{g_{\mu\nu}\frac{dx^\mu}{d\lambda}\frac{dx^\nu}{d\lambda}}d\lambda$$

has the geodesic equation as equation of motion. This result is only due to the form of the connection as depending only on derivatives of the metric tensor. This also applies to Einstein-Cartan theory thanks to the fact that ${\Gamma^\alpha}_{\beta\gamma} \dot x^\beta \dot x^\gamma$ does not depend on any antisymmetric term of the connection.

However, in the case of affine gravity, there are terms that may change the nature of a geodesic from the Levi-Civitta connection. In this case, what is the action of a point particle? I assume that it should still be a geodesic, is there an interaction term with the non-metricity tensor in this case?