The Myers theorem, also known as the Bonnet–Myers theorem, is a classical theorem in Riemannian geometry

Riemannian geometry

Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives, in particular, local notions of angle, length...

. The strong form was proven by Sumner Byron Myers

Sumner Byron Myers

Sumner Byron Myers was an American mathematician specialized in topology. He studied at Harvard University under H. C. Marston Morse, where he was graduated with a Ph.D. in 1932...

. The theorem states that if Ricci curvature

Ricci curvature

In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, represents the amount by which the volume element of a geodesic ball in a curved Riemannian manifold deviates from that of the standard ball in Euclidean space...

 of a complete

Complete space

In mathematical analysis, a metric space M is called complete if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M....

 Riemannian manifold M is bounded below by (n − 1)k > 0, then its diameter is at most π/√k. The weaker result, due to Ossian Bonnet, has the same conclusion but under the stronger assumption that the sectional curvature

Sectional curvature

In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature K depends on a two-dimensional plane σp in the tangent space at p...

s be bounded below by k.

Moreover, if the diameter is equal to π/√k, then the manifold is isometric

Isometry

In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...

 to a sphere of a constant sectional curvature

Sectional curvature

In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature K depends on a two-dimensional plane σp in the tangent space at p...

 k. This rigidity result is due to , and is often known as Cheng's theorem.

This result also holds for the universal cover of such a Riemannian manifold, in particular both M and its cover are compact, so the cover is finite-sheeted and M has finite fundamental group

Fundamental group

In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...

.