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 Riemannian manifold of dimension ${\displaystyle n}$ whose Ricci curvature satisfies ${\displaystyle \operatorname {Ric} (g)\geq (n-1)k}$ for some positive real number ${\displaystyle k.}$ Then any two points of M can be joined by a geodesic segment of length at most ${\textstyle {\frac {\pi }{\sqrt {k}}}{\text{.}}}$

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.