# 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 $$nabla g = 0$$?