On the different ways of determining curvature on manifolds

[trees and plants]

Hello. Why do we have different ways of determining curvature on manifolds like the sectional curvature, the scalar curvature, the Riemann curvature tensor , the Ricci curvature? What are their different uses on manifolds? Do they allow each of them different applications on manifolds? Thank you.

 

[andrewkirk]

The Riemann tensor contains all the information about curvature at a point in a manifold. Curvature at a point can vary in different directions - for instance compare curvature of a cylinder vs a sphere vs a saddle surface embedded in Euclidean 3-space. The wiki article on principal curvatures of a 2D manifold explains that concept nicely.

Since the Riemann tensor applies to a manifold with any number of dimensions, you can imagine it would need a lot of data (to the order of $n^4/2$ real numbers) to encode all the info about curvature of a n-dimensional manifold at a point.

I think of the Riemann tensor loosely as follows. Since it takes four inputs and returns a scalar, we can also consider it as taking three vector inputs - v1, v2 and v3 - and returning a fourth vector v4. If we parallel transport vectors v1, v2 and v3 around a tiny rectangle in the manifold where we first head off in direction v2, then turn into the direction of (parallel transported) v3, then turn in the direction of parallel transported v2, then turn into the direction of parallel transported v3 until we get back to the start point, v4 approximately gives the deviation between the original v1 and its parallel transported version. Every different combination of v1 (vector to be compared with original) and v2, v3 (route for rectangle journey) can give a different answer, so there's a lot of info in that tensor.

The Ricci tensor is a summary of info in the Riemann tensor. Like all summaries, it loses some info. I've never seen an intuitive, physical explanation of the meaning of the Ricci tensor. But I know it is used for calculating Ricci flows, which are useful. In 3D, the Ricci tensor contains all the info in the Riemann tensor. But in higher dimensions, it loses information.

The scalar curvature is a summary of the Ricci tensor, and hence also of the Riemann tensor. As it is a single real number, you can see it is very summarised, and has discarded most of the detail. According to wikipedia, the scalar curvature represents the amount by which the volume of a small geodesic ball in a Riemannian manifold deviates from that of the standard ball in Euclidean space.

Referring back to the wiki article on principal curvatures, note that the Gaussian curvature is a scalar that summarises the richer curvature info provided by the two principal curvatures at a point on a 2D manifold in Euclidean 3-space. That's another case where we use both detailed and summarised (scalar) curvature measures.