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

Question

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 $d(a, b)$.

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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 $\nabla g = 0$?

Answer 1

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.

To further advance in the subject, one could look at this article:

[2] Michael Eastwood, Vladimir S. Matveev, Metric connections in projective differential geometry, arxiv.

On a different side, regarding the case of metric connection with torsion, perhaps the next article is of a certain interest:

[3] Ilka Agricola, Christian Thier, The geodesics of metric connections with vectorial torsion, arxiv.

Maybe, there are other directions, that I am not aware of.