Two metrics $m_1$ and $m_2$ on a manifold $M$ are conformally equivalent if $m_2 = f \,m_1$ for $f$ a smooth positive function. In this lecture we investigate how the Levi-Civita connection and its curvature tensor change when we alter the metric via a conformal equivalence. We then state (but sadly don’t have time to prove) the Killing-Hopf Theorem which shows that if $(M^n,m)$ is any complete, connected and simply connected Riemannian manifold of constant curvature $ \kappa$ then there is an isometry between $(M,m)$ and exactly one of

  • $\mathbb{R}^n$ with the flat metric (if $ \kappa = 0$).
  • The sphere $S^n$ with the round metric (if $ \kappa > 0$),
  • The hyperbolic plane $ \mathbb{H}^n$ 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 $m$ such that the Ricci curvature and the metric differ only by a constant factor:

$$ \operatorname{Ric}_m = \lambda \, m, \qquad \lambda \in \mathbb{R}.$$

We conclude by giving three (slightly wishy-washy) reasons as to why Einstein metrics are the “best” sort of metric. 🤓

Comments and questions?