Is parallel transport or a connection needed for geodesic computation?

Question

I have been reading a bit more about differential geometry and I'm interested in it from the practical computational perspective. I have seen some places where it is mentioned that you cannot move between different tangent spaces to another without a connection ($\nabla$), hence the need for Christoffel symbols to define the geodesic equation. However, I saw some other practical implementations that seem to imply that no connection (at least explicitly?) is needed to compute geodesic, where you basically have a parametrized curve $c(t)$ and you minimize an energy functional (which seems related to Dirichlet energy) using the metric tensor field across this curve and you get a geodesic (or "one" geodesic given non-uniqueness). So my question is: if you minimize energy using the metric tensor for the inner product, where does the connection or parallel transport fits there ? Is it even needed at all for the geodesic curve computation (or estimation) ?

Answer 1

Let $M$ be a manifold such that there is a length function, where each smooth curve $c: [a,b] \rightarrow M$ has a length $\ell[c]$ and $\ell$ satisfies some expected geometric properties. Then you can define a distance function between two points as the infimum of lengths of curves connecting the two points. A geodesic is now a curve such that given any two sufficiently close points, the length of the curve segment connecting the two points is equal to the distance between the two points. Lots of things are hard to do in this setting. Even the existence of any geodesics is in question.

So far, no connection is needed. It is possible to proceed further without a connection. However, the above can be done within the context of Riemannian geometry, and it turns out that a geodesic, with respect to a constant speed parameterization satisfies the geodesic equation $$ \nabla_{c'}c' = 0. $$ Equivalently, the velocity vector $c'$ is parallel along the geodesic $c$.

This equation, however, does not use the Riemannian metric itself, only the Levi-Civita connection. It follows that if you have a connection on a manifold, then you can define geodesics using the geodesic equation, even if the connection is not the Levi-Civita connection of a Riemannian metric. In particular, no distance or length function is used for this definition of a geodesic. This approach is important in general relativity for defining null geodesics.