Myers's theorem#Corollaries

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

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 Einstein manifold 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 {{math|π^*g}} on $N.$ Since $\pi$ is a local isometry, Myers' theorem applies to the Riemannian manifold {{math|(N,π^*g)}} 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 {{math|d_g(p,q) ≤ π/{{radic|k}}}}. In 1975, Shiu-Yuen Cheng proved:

{{quote|Let $(M, g)$ be a complete and smooth Riemannian manifold of dimension {{mvar|n}}. If {{mvar|k}} is a positive number with {{math|Ric^g ≥ (n-1)k}}, and if there exists {{mvar|p}} and {{mvar|q}} in {{mvar|M}} with {{math|d_g(p,q) {{=}} π/{{radic|k}}}}, then {{math|(M,g)}} is simply-connected and has constant sectional curvature {{mvar|k}}.}}

References

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=289–297}}
{{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=401–404}}

Category:Geometric inequalities

Category:Theorems in Riemannian geometry

Myers's theorem#Corollaries

Corollaries

Cheng's diameter rigidity theorem

See also

References