Myers theorem

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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 (n − 1)k > 0, then its diameter is at most π/√k.

Moreover, if the diameter is equal to π/√k, then the manifold is isometric to a sphere of a constant sectional curvature k.

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.

[edit] 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)