In Euclidean geometry, directions are well-defined because since the curvature vanishes, tangent spaces are canonically isometric via the holonomy of Levi-Civitta. This ceases to be true in non-zero curvature. What I mean by that is that it really makes sense to compare directions at any two points in Euclidean geometry, whereas it does not in other geometries.

what is a direction in a non-Euclidean geometry ?

The curves I work with in my comment do not depend on the metric (as in the question).

Thanks for the references. Maybe I didn't insist enough on the fact that I want to work with isotopy classes of metrics, in which case we know that there exists at least one finite collection of homotopy classes whose lengths completely determine the metric (such a set can be made of cardinality 9g - 9 if I remember correctly).