"Visualizing" Ricci curvature Can someone help me visualize the Ricci curvature? Since it is easier to visualize a surface bending in 3-D, let's try to view this as a sheet with one spatial dimension and one time dimension and embedding into euclidean 3-D. Since the metric can always be written locally as ds^2 = -dt^2 + dx^2, deviations to this must be quadratic or higher orders, and these quadratic terms are the "curvature" pieces, right? The examples I know are: Positive curvature would be like the top of a sphere. Negative curvature would be like a hyperbolic sheet. Now if we look at the Ricci tensor, what would be an example of something that has different "curvature" in different "directions"? The hyperbolic sheet already kind of looks like that to me, but that is constant curvature. So I'm having trouble visualizing it. Any help? Are there some good visual illustrations from websites or books that people can recommend? For a concrete example to help me visualize: Consider a spherical shell, with a large mass attached to the north and south poles. There is a lot of symmetry here. We should be able to use that to argue the form of the Ricci tensor at the origin. I want to be able to visualize curvature well enough that I can understand how to make such symmetry arguments myself.