In this lecture we define three new notions of curvature.

The first is the sectional curvature. This is a much easier gadget than the full curvature tensor. 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 $TM$. We prove that the sectional curvature completely determines the full curvature tensor.

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

We state (but sadly don’t have time to prove) the Killing-Hopf Theorem, which says that if $(M,g)$ is any complete, connected and simply connected Riemannian manifold of constant curvature $\kappa$ then $(M,g)$ is isometric to exactly one of:

• $\mathbb{R}^m$ with the flat metric (if $\kappa = 0$).
• The sphere $S^m$ with the round metric (if $\kappa > 0$),
• The hyperbolic plane $\mathbb{H}^m$ with the hyperbolic metric (if $\kappa <0$).

After this we move onto Ricci curvature, which is a $(0,2)$ tensor obtained by taking the trace of the Riemannian curvature. This leads us to the definition of an Einstein metric, which is a metric $g$ such that the Ricci curvature and the metric differ only by a constant factor:
$$\operatorname{Ric}_g = \lambda , g, \qquad \lambda \in \mathbb{R}.$$

We then prove (two versions of) Schur’s Theorem, which illustrate a key difference between Riemannian geometry in dimension 2 and in dimensions $m \ge 3$.

Finally, we introduce the scalar curvature, which is the trace of the Ricci curvature.

##### 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.