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 and connected Riemannian manifold of dimension ${\displaystyle n}$ whose Ricci curvature satisfies for some fixed positive real number ${\displaystyle r}$ the inequality ${\displaystyle \operatorname {Ric} _{p}(v)\geq (n-1){\frac {1}{r^{2}}}}$ for every ${\displaystyle p\in M}$ and ${\displaystyle v\in T_{p}M}$ unitary. Then any two points of M can be joined by a geodesic segment of length at most ${\displaystyle \pi r}$.

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 ${\displaystyle (M,g)}$ is finite. The Hopf-Rinow theorem therefore implies that ${\displaystyle M}$ must be compact, as a closed (and hence compact) ball of radius ${\displaystyle \pi /{\sqrt {k}}}$ in any tangent space is carried onto all of ${\displaystyle M}$ 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 ${\displaystyle M}$ is connected, there exists the smooth universal covering map ${\displaystyle \pi :N\to M.}$ One may consider the pull-back metric π^*g on ${\displaystyle N.}$ Since ${\displaystyle \pi }$ is a local isometry, Myers' theorem applies to the Riemannian manifold (N,π^*g) and hence ${\displaystyle N}$ is compact and the covering map is finite. This implies that the fundamental group of ${\displaystyle M}$ is finite.

Cheng's diameter rigidity theorem

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

Let ${\displaystyle (M,g)}$ 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) – 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, 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