Editing Myers's theorem

{{Short description|Bounds the length of geodetic segments in Riemannian manifolds based in Ricci curvature}}
'''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:

{{block indent|1= Let <math>(M, g)</math> be a complete and connected Riemannian manifold of dimension <math>n</math> whose [[Ricci curvature]] satisfies for some fixed positive real number <math>r</math> the inequality <math>\operatorname{Ric}_{p}(v)\geq (n-1)\frac{1}{r^2}</math> for every <math>p\in M</math> and <math>v\in T_{p}M</math> of unit length. Then any two points of ''M'' can be joined by a geodesic segment of length at most <math>\pi r</math>.}}

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 a set|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|π<sup>*</sup>''g''}} on <math>N.</math> Since <math>\pi</math> is a local isometry, Myers' theorem applies to the Riemannian manifold {{math|(''N'',π<sup>*</sup>''g'')}} 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''<sub>''g''</sub>(''p'',''q'') ≤ ''π''/{{radic|''k''}}}}. In 1975, [[Shiu-Yuen Cheng]] proved:
{{quote|Let <math>(M, g)</math> be a complete and smooth Riemannian manifold of dimension {{mvar|n}}. If {{mvar|k}} is a positive number with {{math|Ric<sup>''g''</sup> ≥ (''n''-1)''k''}}, and if there exists {{mvar|p}} and {{mvar|q}} in {{mvar|M}} with {{math|''d''<sub>''g''</sub>(''p'',''q'') {{=}} ''π''/{{radic|''k''}}}}, then {{math|(''M'',''g'')}} is simply-connected and has constant [[sectional curvature]] {{mvar|k}}.}}

== See also ==

* {{annotated link|Gromov's compactness theorem (geometry)}}

==References==

{{reflist}}
{{reflist|group=note}}

* Ambrose, W. A theorem of Myers. Duke Math. J. 24 (1957), 345–348.
* {{Citation|doi=10.1007/BF01214381|last1=Cheng|first1=Shiu Yuen|title=Eigenvalue comparison theorems and its geometric applications|mr=0378001|year=1975|journal=[[Mathematische Zeitschrift]]|issn=0025-5874|volume=143|issue=3|pages=289&ndash;297}}
* {{citation|first=M. P.|last=do Carmo|authorlink=Manfredo do Carmo|title=Riemannian Geometry|publisher=Birkhäuser|publication-place=Boston, Mass.|year=1992|isbn=0-8176-3490-8 }}
* {{citation|doi=10.1215/S0012-7094-41-00832-3|first=S. B.|last=Myers|title=Riemannian manifolds with positive mean curvature|journal=Duke Mathematical Journal|volume=8|issue=2|year=1941|pages=401&ndash;404}}

{{Riemannian geometry}}
{{Manifolds}}

[[Category:Geometric inequalities]]
[[Category:Theorems in Riemannian geometry]]