A formula I know for the number of functionally independent components of the curvature tensor is: (n^2)(n^2 -1)/12. It gives 1 for n=2, 6 for n=2, 20 for n=4. However, for a metric space (with symmetric metric), the curvature tensor is completely specified by the metric tensor. For n=4, there are only 10 different components of the metric, and one can argue there are only 6 functions of the manifold needed to specify the geometry, as all metric representations connected by diffeomorphism represent the same geometry. How does one square this with 20 functionally independent components of the curvature tensor?