Example Application Ricci Tensor & Scalar for 3D Understanding

[space-time]

Can anyone show me an example of applying the Ricci curvature tensor to something other than GR? I also ask the same for the curvature scalar. Lately I've been trying to truly increase my understanding of curvature, so that I can see exactly how solutions of the EFE's predict the existence and presence of things like wormholes, black holes, and closed timelike curves. While I do quantitatively understand how to derive the curvature, my qualitative understanding of what this curvature truly is and what it actually describes could use some polishing. Perhaps if I see it applied to something else (like something in 3D instead of the 4D of relativity) then maybe I will understand the significance of all the terms of the Ricci tensor, the tensor itself, and the curvature scalar.

Thank you.

 

[bcrowell]

In two dimensions, all of these measures of curvature are basically the Gaussian curvature, which has a lot of nice applications. For example, it explains why you can fold a piece of pizza that you're eating to keep it from drooping.