Myers's theorem

Myers's theorem, also known as the Bonnet–Myers theorem, is a celebrated, fundamental theorem in the mathematical field of Riemannian geometry. It was discovered by Sumner Byron Myers in 1941. It asserts the following:

Let ${\displaystyle (M,g)}$ be a complete and connected Riemannian manifold of dimension ${\displaystyle n}$ whose Ricci curvature satisfies for some fixed positive real number ${\displaystyle r}$ the inequality ${\displaystyle \operatorname {Ric} _{p}(v)\geq (n-1){\frac {1}{r^{2}}}}$ for every ${\displaystyle p\in M}$ and ${\displaystyle v\in T_{p}M}$ of unit length. Then any two points of M can be joined by a geodesic segment of length at most ${\displaystyle \pi r}$.

In the special case of surfaces, this result was proved by Ossian Bonnet in 1855. For a surface, the Gauss, sectional, and Ricci curvatures are all the same, but Bonnet's proof easily generalizes to higher dimensions if one assumes a positive lower bound on the sectional curvature. Myers' key contribution was therefore to show that a Ricci lower bound is all that is needed to reach the same conclusion.