I am considering the Schwarzschild metric. I have calculated my Christoffel symbols and am able to calculate the Riemann tensor (I think). In short, I have done a bunch of work to find this thing called the Riemann curvature tensor. As I understand it, if I take a vector $Z$ at a point on my manifold, $R(X,Y)Z$ gives me the difference between my original vector and the parallel transported one. But I really don't know what to do next. I want to use this tensor to compute some useful piece of physical information. For instance, I can use $$v_f = v_i +at$$ to calculate the final velocity of an object dropped for a duration $t$ and an acceleration $a = g$. This is an example of an application that yields interesting/useful information. I want something that allows me to do something similarly interesting with my curvature tensor.
Can someone provide me with an example of what I can do with my curvature tensor? What's an example of a useful application?
EDIT: On page 118 of Wald, he says
However, the Schwarzschild solution, which describes the exact exterior field of spherical body, predicts tiny departures from Newtonian theory for the motion of planets in our solar system, and, in addition, predict the "bending of light", the gravitational redshift of light, and "time delay" effects. These four predictions have been accurately confirmed by precise measurements.
Thus, I assume then that one of those four examples must require use of the Riemann tensor somehow. Perhaps someone can explain which one(s) do and how I might go about using my curvature tensor to describe these phenomena.