Does the existence of the Levi-Civita connection depend on whether or not we define a metric on our smooth manifold? 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.

I think maybe you're misinterpreting the phrase "the Levi-Civita connection." Despite the definite article, there isn't just one such connection -- every Riemannian metric has its own Levi-Civita connection, uniquely determined by the metric. So the question in your title doesn't make sense -- without a specific choice of metric, "the Levi-Civita connection" has no meaning. It's like asking if "the derivative" exists without specifying a particular function to take the derivative of.
