Myers's theorem

It was discovered by Sumner Byron Myers in 1941.

It asserts the following: 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.

As a very particular case, this shows that any complete and noncompact smooth Einstein manifold must have nonpositive Einstein constant.

is connected, there exists the smooth universal covering map

is a local isometry, Myers' theorem applies to the Riemannian manifold (N,π*g) and hence

is compact and the covering map is finite.

be a complete and smooth Riemannian manifold of dimension n. If k is a positive number with Ricg ≥ (n-1)k, and if there exists p and q in M with dg(p,q) = π/√k, then (M,g) is simply-connected and has constant sectional curvature k.