Purpose of sectional curvature

Question

I'm trying to get more knowledge on Riemann, Ricci and Einstein tensors and while reading Differential Geometry Curves Surfaces Manifolds Third Edition - W. Kühnel I came across the sectional curvature of the Riemann manifold with respect to the plane $\sigma$. I don't really see what the purpose of this curvature is, since I feel like the Riemann tensor (curvature tensor) is perfectly capable to give us enough information on the curvature of a manifold. So could anyone explain me what the exact purpose of the sectional curvature is?

Answer 1

The Riemann curvature indeed contains all information. The other way around as well, you can reconstruct the Riemann curvature from the sectional curvature.

One problem with the Riemann curvature is that it is very abstract. It contains a lot of information about the manifold and its Riemannian structure, but not always in a very straightforward way. Arguably the only intuitively clear concept is the Gaussian curvature. The sectional curvature allows you to reduce the Riemann curvature to the Gaussian curvature of all (geodesic) surfaces through the point in question.

The (or rather a) technical definition, starting from the curvature, is as follows. Let $g$ be the metric, and $R$ the curvature, as a map $R:\Lambda^2TM\to\operatorname{End}TM$. Then the expression $g(R(X,Y)Z, W)$ for $X,Y,Z,W\in\Gamma(TM)$ is

• alternating in both $X,Y$ and in $Z,W$
• symmetric in the pairs $(X,Y)$ and $(Z,W)$.

In other words, it defines a symmetric bilinear form on $\Lambda^2TM$. Since a symmetric bilinear form is fully determined by its values on the diagonal, the sectional curvature, which is by definition the restriction of this form to its diagonal, fully determines the Riemann curvature tensor.

When defined in this way, the equivalence is clear, but the intuition is still far. However, there is a very intuitive approach to the sectional curvature as well: any pair of tangent vectors at a point can locally be integrated (by means of the exponential map) to a geodetic surface. The sectional curvature for that pair of tangent vectors (or for their exterior product) is the Gaussian curvature of that surface.

There is another intuitive interpretation of the sectional curvature that is nice and practically useful: let $u,v\in T_xM$ be orthogonal unit vectors. The sectional curvature $\kappa(u\wedge v)$ is the lowest order, i.e. quadratic, infinitesimal correction to the distance between the geodesics $\gamma_u,\gamma_v$ where $\gamma_u(t) := \exp_x(tu)$:

$$d\left(\gamma_u(t),\gamma_v(t)\right) \equiv \sqrt2 t\left(1 - \frac\kappa{12}t^2 + \mathcal O(t^3)\right)\ \ \ \ (t\to0).$$

In particular we see that positive sectional curvature reduces distances w.r.t. flat and negative sectional curvature increases it.

Answer 2

Here's an example:

A Riemannian manifold $(M,g)$ (I also understand pseudo-Riemannian manifolds under the term "Riemannian") is a constant curvature space if $$ \mathrm{Sec}(\Pi)=c $$for the same number $c\in\mathbb R$ for all 2-planes $\Pi$ (all non-isotropic 2-planes, if we are also considering indefinite metrics) at all points of $M$, where $\mathrm{Sec}$ is the sectional curvature.

Constant curvature spaces are pretty important, they are basically the "most symmetric" Riemannian spaces.

On the other hand if $R$ denotes the Riemann curvature tensor, it doesn't really make sense to talk about $R$ being constant.

The closest notion is that $\nabla R=0$, i.e. the curvature tensor is parallel. But this is actually inequivalent to $\mathrm{Sec}=\mathrm{const}$. A constant curvature space will always satisfy $\nabla R=0$, but a space with $\nabla R=0$ is called a locally symmetric space, and is a strictly weaker notion than constant curvature spaces. There are Riemannian manifolds which are locally symmetric but not constant curvature.

Constant curvature spaces can also be formulated in terms of the curvature tensor btw, a space is constant curvature if and only if$$ R_{ijkl}=k(g_{ik}g_{jl}-g_{il}g_{jk}) $$for some constant $k\in\mathbb R$, but this is a far more unintuitive notion that $\mathrm{Sec}$ being constant.

So this is an example where something can be formulated much more easily via the sectional curvature than the curvature tensor, even though the two contain the same data.

Also from a conceptional point of view, the sectional curvature is a direct generalization of Gaussian curvature, while the curvature tensor is a bit less immediately related object.