Myers's theorem explained

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:

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 on N. Since \pi is a local isometry, Myers' theorem applies to the Riemannian manifold 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 . In 1975, Shiu-Yuen Cheng proved:

References

• Ambrose, W. A theorem of Myers. Duke Math. J. 24 (1957), 345–348.