Myers's theorem, also known as the Bonnet–Myers theorem, is a celebrated, fundamental theorem in the mathematical field of

Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as manifold, smooth manifolds with a ''Riemannian metric'' (an inner product on the tangent space at each point that varies smooth function, smo ...

. It was discovered by

Sumner Byron Myers Sumner Byron Myers (February 19, 1910 – October 8, 1955) was an American mathematician specializing in topology and differential geometry. He studied at Harvard University under H. C. Marston Morse, Tucker, A: Interview with Albert Tucker'', Pr ...

in 1941. It asserts the following: In the special case of surfaces, this result was proved by

Ossian Bonnet Pierre Ossian Bonnet (; 22 December 1819, Montpellier – 22 June 1892, Paris) was a French mathematician. He made some important contributions to the differential geometry of surfaces, including the Gauss–Bonnet theorem. Biography Early year ...

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 In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature ''K''(σ''p'') depends on a two-dimensional linear subspace σ''p'' of the tangent space at a po ...

. 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 In geometry, a diameter of a circle is any straight line segment that passes through the centre of the circle and whose endpoints lie on the circle. It can also be defined as the longest Chord (geometry), chord of the circle. Both definitions a ...

(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

Einstein manifold In differential geometry and mathematical physics, an Einstein manifold is a Riemannian or pseudo-Riemannian differentiable manifold whose Ricci tensor is proportional to the metric. They are named after Albert Einstein because this condition is ...

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 Shiu-Yuen Cheng (鄭紹遠) is a Hong Kong mathematician. He is currently the Chair Professor of Mathematics at the Hong Kong University of Science and Technology. Cheng received his Ph.D. in 1974, under the supervision of Shiing-Shen Chern, from ...

proved:

References

* Ambrose, W. A theorem of Myers. Duke Math. J. 24 (1957), 345–348. * * * {{Manifolds Geometric inequalities Theorems in Riemannian geometry

Corollaries

Cheng's diameter rigidity theorem

See also

References