Myers's theorem
Bounds the length of geodetic segments in Riemannian manifolds based in Ricci curvature / From Wikipedia, the free encyclopedia
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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:
Let be a complete and connected Riemannian manifold of dimension whose Ricci curvature satisfies for some fixed positive real number the inequality for every and unitary. Then any two points of M can be joined by a geodesic segment of length at most .
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.