The Christoffel symbols of the Levi-Civita connection are calculated through the metric, but does that necessarily mean that its existence depends on whether or not we have a metric?
Specifically, the Levi-Civita connection offers a specific way to parallel transport a vector along a manifold, so if we don't have a metric on a manifold, does this particular way to parallel transport a vector along the manifold gets "lost"? I mean, of course we need the metric to determine the Levi-Civita connection, but as a geometrical concept, it seems intuitive to me that as a way to transport vectors, its existence should not depend on whether on not we defined a metric on our manifold.
Note that this question is motivated by the fact that we define connections before even talking about a metric. But in my Riemannian geometry course, we defined the Levi-Civita connection through the metric, but I wanted to know if this necessarily means that it can't be defined(as a concept) without a metric.
Thanks in advance.