Myers's theorem

Remarks

The idea of the proof: let $\gamma$ be a minimizing unit-speed geodesic of length $L > \pi/\sqrt{k}$. Take an orthonormal frame $E_1, \dots, E_{n-1}$ of parallel fields normal to $\gamma$ and the variation fields $\sin(\pi t / L) E_i$. By the second variation formula, the sum of the second variations of length is negative, so one of them is negative and $\gamma$ is not minimizing, a contradiction. Part 2 follows from part 1 and the Hopf–Rinow theorem. For part 3, apply parts 1 and 2 to the universal cover with the pulled-back metric: it is compact, so the covering has finitely many sheets.

Example: the unit sphere $S^n$ has $\operatorname{Ric} = (n-1) g$ and diameter exactly $\pi$, so the bound is sharp. By a theorem of Cheng, equality of the diameter with $\pi/\sqrt{k}$ holds only for the round sphere of radius $1/\sqrt{k}$.

The uniform lower bound is needed: the paraboloid $z = x^2 + y^2$ in $\R^3$ is complete with positive curvature, but its curvature tends to $0$ at infinity and it is not compact.

Going deeper

For a student

Myers's theorem is one of the first results linking a lower bound on Ricci curvature to global topology. For example, it shows that a compact manifold with infinite fundamental group, such as a torus, carries no metric of positive Ricci curvature. It has a sharp volume analogue, the Bishop–Gromov inequality, and it is used in general relativity to show that certain spacetimes are closed.