Myers's theorem

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. It asserts the following:

Let $(M,g)$ be a complete and smooth Riemannian manifold of dimension $n$ whose Ricci curvature satisfies $Ric g \geq (n -1) k$ for some positive number $k$ . Then any two points of M can be joined by a geodesic segment of length $\leq \pi /{\sqrt {k}}$ .

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 $(M, g)$ 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 $π : N \to M$ . 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 $d g (p, q) \leq π / \sqrt k$ . 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 $Ric g \geq (n -1) k$ , and if there exists $p$ and $q$ in $M$ with $d g (p, q) = π / \sqrt k$ , then $(M, g)$ is simply-connected and has constant sectional curvature $k$ .

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

Corollaries

Cheng's diameter rigidity theorem

See also

References