Geodesics on a Riemannian variety under non-Levi-Civita connections

I am a beginner on this subject. Please clarify if something is ambiguous, unclear or false. In particular, I try to understand how to think geodesics in arbitrary connections.

Usually, when we talk about geodesics on a Riemannian multiple $ (M, g) $, we talk about geodesic regarding the Levi-Civita connection.

We can also talk about a metric on connected $ (M, g) $ induced by the metric $ g $ given by $$
d (a, b) = inf left { int_ gamma g ( gamma (t), gamma (t)) , dt: gamma (0) = a, gamma (1) = b right }.
$$

In particular, if $ (M, g) $ complete geodetic, there is a geodesic curve with respect to the Levi-Civita connection of $ a $ at $ b $ length $ d (a, b) $.


What is the relationship between geodesics and metrics for an arbitrary connection $ nabla $?

In particular, is there a geodesic curve with respect to $ nabla $ of $ a $ at $ b $ length $ d (a, b) $?

And what would happen if $ nabla $ meets the first condition of a Levi-Civita connection, namely $ nabla g = $ 0?