I am confused in one question in general relativity, why we can always express a space-time geometry only by metric. It means a metric, which is just about distance in tangent space, can tell us all the information about the manifold.
I know there are standard proofs, for instance, we can express connection by metric, and so the Riemann curvature. However, I am not very satisfied by these answers. I still want a more direct reason for that.
To my understanding, a metric just defines the distance, the length of tangent vectors, however, Riemann curvature, in my eyes, tell us more, for instance, how a line differ from a direct line and how a vector travel along a closed path.
I believe there must be some neat and beautiful argument show that metric is enough, is everything.
This question is quite vague, so please just feel like a chatting.