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Myers's theorem

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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 of unit length. 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.

Corollaries

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The conclusion of the theorem says, in particular, that the diameter of is finite. Therefore must be compact, as a closed (and hence compact) ball of finite radius in any tangent space is carried onto all of by the exponential map.

As a very particular case, this shows that any complete and noncompact smooth Riemannian manifold which is Einstein must have nonpositive Einstein constant.

Since is connected, there exists the smooth universal covering map One may consider the pull-back metric π*g on Since is a local isometry, Myers' theorem applies to the Riemannian manifold (N*g) and hence is compact and the covering map is finite. This implies that the fundamental group of is finite.

Cheng's diameter rigidity theorem

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The conclusion of Myers' theorem says that for any one has dg(p,q) ≤ π/k. In 1975, Shiu-Yuen Cheng proved:

Let be a complete and smooth Riemannian manifold of dimension n. If k is a positive number with Ricg ≥ (n-1)k, and if there exists p and q in M with dg(p,q) = π/k, then (M,g) is simply-connected and has constant sectional curvature k.

See also

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References

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  • Ambrose, W. A theorem of Myers. Duke Math. J. 24 (1957), 345–348.
  • Cheng, Shiu Yuen (1975), "Eigenvalue comparison theorems and its geometric applications", Mathematische Zeitschrift, 143 (3): 289–297, doi:10.1007/BF01214381, ISSN 0025-5874, MR 0378001
  • do Carmo, M. P. (1992), Riemannian Geometry, Boston, Mass.: Birkhäuser, ISBN 0-8176-3490-8
  • Myers, S. B. (1941), "Riemannian manifolds with positive mean curvature", Duke Mathematical Journal, 8 (2): 401–404, doi:10.1215/S0012-7094-41-00832-3