Editing Myers's theorem (section)

{{Short description|Bounds the length of geodetic segments in Riemannian manifolds based in Ricci curvature}}
'''Myers's theorem''', also known as the '''Bonnet–Myers theorem''', is a celebrated, fundamental  theorem in the mathematical field of [[Riemannian geometry]].  It was discovered by [[Sumner Byron Myers]] in 1941. It asserts the following:

{{block indent|1= Let <math>(M, g)</math> be a complete and connected Riemannian manifold of dimension <math>n</math> whose [[Ricci curvature]] satisfies for some fixed positive real number <math>r</math> the inequality <math>\operatorname{Ric}_{p}(v)\geq (n-1)\frac{1}{r^2}</math> for every <math>p\in M</math> and <math>v\in T_{p}M</math> of unit length. Then any two points of ''M'' can be joined by a geodesic segment of length at most <math>\pi r</math>.}}

In the special case of surfaces, this result was proved by [[Ossian Bonnet]] in 1855. For a surface, the  Gauss, sectional, and Ricci curvatures are all the same, but Bonnet's proof easily generalizes to higher dimensions if one assumes a positive lower bound on the  [[sectional curvature]]. Myers' key contribution was therefore to show that a Ricci lower bound is all that is needed to reach the same conclusion.