A manifold admits a huge plurality of connections. If I have a metric (and a few other reasonable preferences) I get a unique metric connection. But if I start with an arbitrary connection, can I find a metric that it could have come from? Why or why not? What implications might this have for differential geometry?
Edit: the visual that I'd had in my head up to now, to make sense of manifolds having many many connections, was of embedding the manifold into Rn and then "squeezing" it: each connection would correspond somehow to a different embedding. It was later I realized (I think) that this implicitly assumes every connection is a metric connection. But, does it really assume that? Is there any sense in which the foregoing is true? (Clearly we are talking about embeddable manifolds, so... second-countable and Hausdorf, I believe).