The Bonnet-Myers Theorem, also known as Myers's Theorem, is a fundamental result of Riemannian geometry discovered by Sumner Byron Myers in 1941, building on an earlier special case due to Ossian Bonnet. It states that a complete Riemannian manifold whose Ricci curvature is bounded below by a fixed positive constant must be compact, with a diameter no larger than a bound determined by that constant, and consequently must have a finite fundamental group. The theorem is one of the classical global results relating a lower curvature bound to strong topological restrictions on the manifold as a whole.
Facts
StatementIf a complete and connected Riemannian manifold of dimension n has Ricci curvature bounded below, for some fixed positive real number r, by (n-1) over r squared in every direction at every point, then any two points of the manifold can be joined by a geodesic segment of length at most pi times r. 1 Classification
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Source Myers's theorem (Wikipedia)
Proved By
Source Myers's theorem (Wikipedia)
Source Myers's theorem (Wikipedia)
Sources
1. Myers's theorem, Wikipedia
Statement section
Let (M,g) be a complete and connected Riemannian manifold of dimension n whose Ricci curvature satisfies for some fixed positive real number r the inequality Ric_p(v) >= (n-1)1/r^2 for every p in M and v in T_pM of unit length. Then any two points of M can be joined by a geodesic segment of length at most pi r.
History section
It was discovered by Sumner Byron Myers in 1941.
View the SourceMyers's theorem (Wikipedia)
- In Branch: Differential Geometry, Lead sentence
Proved By: Sumner Byron Myers, Lead sentence, attribution
discovered by Sumner Byron Myers in 1941
Proved By: Pierre Ossian Bonnet, Lead sentence, attribution
proved by Ossian Bonnet in 1855
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