In this lecture we first introduce a $(0,4)$ version of the curvature tensor[1] of a connection $ \nabla$ on a Riemannian manifold $(M,m)$, which we denote the $ \mathcal{R}^{ \nabla}_m$. We show that $ \mathcal{R}^{ \nabla}_m$ satisfies an additional “bonus” symmetry if $ \nabla$ is the Levi-Civita connection of $m$, namely: $$ \mathcal{R}^{ \nabla}_m(W,Z,X,Y) = \mathcal{R}^{\nabla}_m(X,Y,W,Z).$$

We then move onto the sectional curvature. This is a much easier gadget than the full curvature tensor—indeed, when $M$ is two-dimensional the sectional curvature reduces to a single smooth function on $M$ (which is then often called the Gaussian curvature). In general one can think of the sectional curvature as a function on the Grassmannian manifold of $2$-planes in $TM$.

Amazingly however the full curvature tensor is completely determined by the sectional curvatures. This is a simple algebraic argument that uses the “bonus” symmetry of $ \mathcal{R}^{\nabla}_m$ alluded to above.

We then introduce spaces of constant curvature, which are spaces for which the sectional curvature is identically constant. For example, the sphere $S^n$ equipped with its standard round metric is a space of constant curvature +1.

Finally we prove Schur’s Theorem, which states that if $M$ is connected and has dimension at least 3, then if the sectional curvatures only depend on the point $x$ and not on the 2-plane $ \Pi \subset T_xM$, then we are necessarily on a space of constant curvature. This illustrates a key difference between Riemannian geometry in dimension 2 and in dimension $n \ge 3$.

Remark

Schur’s Theorem is named after the German mathematician Axel Schur. There are several other theorems in combinatorics and Ramsey theory all also called “Schur’s Theorem”. These however refer to the Russian mathematician Issai Schur.


  1. Beware that many textbooks use different ( = inferior) sign conventions for this version of the curvature tensor. ↩︎

Comments and questions?