How are "scalar curvature" and "sectional curvature" related?

Question

I was browsing wikipedia and was puzzeling about what is the difference between:

"scalar curvature" https://en.wikipedia.org/wiki/Scalar_curvature and "sectional curvature" https://en.wikipedia.org/wiki/Sectional_curvature ?

For 2-dimensional surfaces they both describe the "Gaussian curvature" https://en.wikipedia.org/wiki/Gaussian_curvature

(From the scalar curvature page "the scalar curvature is twice the Gaussian curvature")

So that made me wonder, they both seem to describe the same thing (curvature of a manifold ) but how are they related, and how can you calculate one from the other?

Also I was editing the page on hyperbolic geometry on wikipedia https://en.wikipedia.org/wiki/Hyperbolic_geometry and was wondering to which of the three curvatures I should refer.

For the two dimensional case I can safely refer to the Gaussian curvature , but for higher dinensional cases which curvature is correct/best?

PS Under similar question I found Relationship beween Ricci curvature and sectional curvature, but am not sure if that makes this question a duplicate (if anything that question is about positive curvature, while mine is about negative curvature)

Answer 1

An important class of problems in Riemannian geometry is to understand the interaction between the curvature and topology.

An example of such interaction is given by the Gauss-Bonnet theorem which relates the geodesic curvature, the Gaussian curvature to the Euler characteristic of a regular surface of class $C^3$.

When studying the geometry of a smooth manifold we need to introduce the commutator of twice covariant differentiating vector fields which is called Riemannian curvature tensor.

As a matter of fact, if in Euclidean space we can change the order of differentiation, on a Riemannian manifold, the Riemann curvature tensor is in general nonzero.

For surfaces, the Riemann curvature tensor is equivalent to the Gaussian curvature K, which is scalar function.

On the other hand in dimensions larger than two, the Riemann curvature tensor is a tensor–field.

Now there are several curvatures associated to the Riemann curvature tensor.

Given a point $p \in M^n$ and two dimensional plane $\Pi$ in the tangent space of M at p, we can define a surface S in M to be the union of all geodesics passing through p and tangent to $\Pi$.

In a neighborhood of p, S is a smooth 2D submanifold of M. Then it is possible to define the sectional curvature $K(\Pi)$ of the 2D plane to be the Gaussian curvature of S at p.

Thus the sectional curvature $K$ of a Riemannian manifold associates to each 2D plane in a tangent space a real number.

You can imagine the sectional curvature as a kind of generalization of the Gaussian curvature.