Myers theorem

Myers theorem

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.

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)


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