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

(M,g)

be a complete and connected Riemannian manifold of dimension

n

whose Ricci curvature satisfies for some fixed positive real number

r

the inequality

\operatorname {Ric} _{p}(v)\geq (n-1){\frac {1}{r^{2}}}

for every

p\in M

and

v\in T_{p}M

of unit length. Then any two points of M can be joined by a geodesic segment of length at most

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

Cheng's diameter rigidity theorem

The conclusion of Myers' theorem says that for any $p,q\in M,$ 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