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 <math>(M, g)</math> is finite. Therefore <math>M</math> must be compact, as a closed (and hence compact) ball of finite radius in any tangent space is carried onto all of <math>M</math> by the exponential map.
As a very particular case, this shows that any complete and noncompact smooth Einstein manifold must have nonpositive Einstein constant.
Since <math>M</math> is connected, there exists the smooth universal covering map <math>\pi : N \to M.</math> One may consider the pull-back metric <math>\pi^*g</math> on <math>N.</math> Since <math>\pi</math> is a local isometry, Myers' theorem applies to the Riemannian manifold <math>(N,\pi^*g)</math> and hence <math>N</math> is compact and the covering map is finite. This implies that the fundamental group of <math>M</math> is finite.
Cheng's diameter rigidity theorem
The conclusion of Myers' theorem says that for any <math>p, q \in M,</math> one has <math>d_g(p,q)\leq\frac{\pi}{\sqrt{k</math>. In 1975, Shiu-Yuen Cheng proved:
</math>, then <math>(M,g)</math> is simply-connected and has constant sectional curvature .
See also
References
- Ambrose, W. A theorem of Myers. Duke Math. J. 24 (1957), 345–348.