Myers's theorem

The Myers 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. {{quote|Let (M,g) be a complete and smooth Riemannian manifold of dimension n. If k is a positive number with Ric^g ≥ (n-1)k, then any two points of M can be joined by a geodesic segment of length ≤ π/√k}.

An earlier result (from 1855), due to Ossian Bonnet, has the same conclusion but under the stronger assumption that the sectional curvatures is bounded below by k.

Corollaries

The conclusion of the theorem says, in particular, that the diameter of (M,g) is finite. The Hopf-Rinow theorem implies that M must be compact, as it is closed and bounded.

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 π : N→M. One may consider the Riemannian metric π^*g on N. Since π is a local diffeomorphism, Myers' theorem applies to the Riemannian manifold (N,π^*g) and hence N is compact. This implies that the fundamental group of M is finite.

Cheng's diameter rigidity theorem

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

Let (M, g) 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)

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, 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