The Myers theorem, also known as the Bonnet-Myers theorem,is a classical theorem in Riemannian geometry. It states that if Ricci curvature of a complete Riemannian manifold M is bounded below by , then its diameter is at most . In mathematics, Riemannian geometry has at least two meanings, one of which is described in this article and another also called elliptic geometry. ... In differential geometry, the Ricci curvature tensor is (0,2)-valent tensor, obtained as a trace of the full curvature tensor. ... In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For...

Moreover, if the diameter is equal to , then the manifold is isometric to a sphere of a constant sectional curvature k. In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces. ... In Riemannian geometry, the sectional curvature is one of the ways to describe the Curvature of Riemannian manifolds. ...

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. In mathematics, specifically topology, a covering map is a continuous surjective map p : C → X, with C and X being topological spaces, which has the following property: to every x in X there exists an open neighborhood U such that p -1(U) is a union of mutually disjoint open... In mathematics, the fundamental group is one of the basic concepts of algebraic topology. ...

References

S. B. Myers, Riemannian manifolds with positive mean curvature, Duke Mathematical Journal Volume 8, Number 2 (1941), 401-404 M. P. do Carmo, Riemannian Geometry, Birkhäuser, Boston, Mass.(1992)