Myers's theorem


The Myers theorem, also known as the Bonnet–Myers theorem, is a classical and well-known theorem in the mathematical field of Riemannian geometry. It was discovered by Sumner Byron Myers in 1941.
Precisely, this says that if is a smooth path of length greater than π/, then there exists and for each a smooth path with and with, with, and such that the associated map is smooth, and such that the length of is less than that of for all. In a topological language, this says that there is a smooth homotopy of with fixed endpoints and which decreases length.
An earlier result, due to Ossian Bonnet, has the same conclusion but under the stronger assumption that the sectional curvatures is bounded below by.

Corollaries

The conclusion of the theorem says, in particular, that the diameter of is finite. The Hopf-Rinow theorem implies that must be compact, as it is closed and bounded.
As a very particular case, this shows that any complete and noncompact smooth Riemannian manifold which is Einstein must have nonpositive Einstein constant.
Consider the smooth universal covering map. One may consider the Riemannian metric on. Since is a local diffeomorphism, Myers' theorem applies to the Riemannian manifold and hence is compact. This implies that the fundamental group of is finite.

Cheng's diameter rigidity theorem

The conclusion of Myers' theorem says that for any and in, one has . In 1975, Shiu-Yuen Cheng proved: