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:

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 therefore implies that $M$ must be compact, as a closed (and hence compact) ball of radius $\pi/\sqrt$ 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.

Consider the smooth universal covering map $\pi : N \to M.$ One may consider the Riemannian metric on $N.$ Since $\pi$ is a local diffeomorphism, Myers' theorem applies to the Riemannian manifold 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 $p, q \in M,$ one has . In 1975, Shiu-Yuen Cheng proved:

See also

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References

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

Differential geometry
Geometric inequalities
Theorems in Riemannian geometry