: Mathematics: Myers's theorem

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Myers's theorem

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 ${\\displaystyle (M,g)}$ be a complete Riemannian manifold of dimension ${\\displaystyle n}$ whose Ricci curvature satisfies ${\\displaystyle \\operatorname {Ric} (g)\\geq (n-1)k}$ for some positive real number ${\\displaystyle k.}$ Then any two points of M can be joined by a geodesic segment of length at most ${\ extstyle {\ rac {\\pi }{\\sqrt {k}}}{\ ext{.}}}$

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
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Myers%27s_theorem)

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