Myers's theorem

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:

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

Corollaries

The conclusion of the theorem says, in particular, that the diameter of [math]\displaystyle{ (M, g) }[/math] is finite. The Hopf-Rinow theorem therefore implies that [math]\displaystyle{ M }[/math] must be compact, as a closed (and hence compact) ball of radius [math]\displaystyle{ \pi/\sqrt{k} }[/math] in any tangent space is carried onto all of [math]\displaystyle{ M }[/math] 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.

Consider the smooth universal covering map [math]\displaystyle{ \pi : N \to M. }[/math] One may consider the Riemannian metric π^*g on [math]\displaystyle{ N. }[/math] Since [math]\displaystyle{ \pi }[/math] is a local diffeomorphism, Myers' theorem applies to the Riemannian manifold (N,π^*g) and hence [math]\displaystyle{ N }[/math] is compact. This implies that the fundamental group of [math]\displaystyle{ M }[/math]is finite.

Cheng's diameter rigidity theorem

The conclusion of Myers' theorem says that for any [math]\displaystyle{ p, q \in M, }[/math] one has d_g(p,q) ≤ π/√k. In 1975, Shiu-Yuen Cheng proved:

Let [math]\displaystyle{ (M, g) }[/math] be a complete and smooth Riemannian manifold of dimension n. If k is a positive number with Ric^g ≥ (n-1)k, and if there exists p and q in M with d_g(p,q) = π/√k, then (M,g) is simply-connected and has constant sectional curvature k.

See also

• Gromov's compactness theorem (geometry) – On when a set of compact Riemannian manifolds of a given dimension is relatively compact

References

• 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 
• 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