# Geodesics on a Riemannian manifold under non-Levi-Civita connections

I'm a beginner on this topic—so please comment if anything is ambiguous, unclear, or wrong. In particular, I'm trying to figure out how to think of geodesics under arbitrary connections.

# Background

Usually, when we talk about geodesics on a Riemannian manifold $$(M, g)$$, we mean geodesics with respect to 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 $$(M,g)$$ is geodesically complete, there exists a geodesic curve with respect to the Levi-Civita connection from $$a$$ to $$b$$ of length $$d(a, b)$$.

# Question

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

In particular, does there exist a geodesic curve with respect to $$\nabla$$ from $$a$$ to $$b$$ of length $$d(a,b)$$?

What if $$\nabla$$ meets the first condition of a Levi-Civita connection, namely $$\nabla g = 0$$?

This question as stated is too broad to have a definite answer, I am afraid.

I would recommend the following article as a good introduction to the subject:

[1] Vladimir S. Matveev, Geodesically equivalent metrics in general relativity, arxiv.