I heard these two statements which don't work together (in my mind):
In 4D spacetime the curvature is encoded within the Riemann tensor. He holds all the information about curvature in spacetime.
The metric describes the intrinsic geometry of a manifold/spacetime, including the curvature.
So, who encodes curvature in 4D spacetime?
I know that Riemann depends on the metric. The Riemann tensor is of rank 4. The metric is only a rank 2 tensor.
Edit: I just looked through my (old) lecture notes and found an explanation I was looking for. "The curvature tensor is a diagnostic tool which tells you if a given metric can be turned into the identity (or the Minkowski metric) for all points in spacetime with some transformation."
The metric can appear do describe some curved thing, but this may be due to the fact that the coordinates were choosen poorly. Given some metric it may be hard to find out if the described space is really curved or just expressed in bad coordinates. The Riemann tensor seems to allow us to figure out quickly if a given space is flat or curved.