Space-time geometry and metric 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.
 A: This is roughly how I think about it. A manifold is defined as a set of dimension $n$ with an open neighborhood at every point that has a 1-1 and continuous map to a Euclidean Space,$E^n$; such that it is locally "like" $E^n$. A single map is a Chart. The set of Charts that cover the entire manifold is an Atlas.(Shutz, Bernard. Geometrical Methods of Mathematical Physics) When the charts are constructed for a set, like a sphere, the metric for the sphere can be derived from the transformation from Euclidean coordinates and the fact that the metric is a tensor. So the metric measures the deviation from Euclidean space. And, if you know exactly how a space differs from a Euclidean space at every point, then you know "everything" about that space. 
But, there are topologically distinct spaces with the same metric -a cylinder is flat. So it doesn't really tell you everything.     
*Edited, to reflect the comment by gns-ank. Replaced an incorrect or imprecise definition of manifold with a rigorous definition.
A: It is a misconception that a Riemannian metric only specifies a norm on the tangent spaces: The inner product also adds the notion of angles and geodesic distance (and thus an actual metric on the manifold).
Personally, I do not find it surprising that distances and angles are enough to specify the geometry - what else should there be? (Torsion, of course - but in general relativity we traditionally ignore this additional degree of freedom...)
