Myers's theorem

Myers's theorem   (Redirected from Myers theorem) Jump to navigation Jump to search 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: Let {displaystyle (M,g)} be a complete Riemannian manifold of dimension {displaystyle n} whose Ricci curvature satisfies {displaystyle Ric^{g}geq (n-1)k} for some positive real number {displaystyle k.} Then any two points of M can be joined by a geodesic segment of length {displaystyle ,leq pi /{sqrt {k}}.} 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.

Contents 1 Corollaries 2 Cheng's diameter rigidity theorem 3 See also 4 References Corollaries The conclusion of the theorem says, in particular, that the diameter of {displaystyle (M,g)} is finite. The Hopf-Rinow theorem therefore implies that {displaystyle M} must be compact, as a closed (and hence compact) ball of radius {displaystyle pi /{sqrt {k}}} in any tangent space is carried onto all of {displaystyle 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 {displaystyle pi :Nto M.} One may consider the Riemannian metric π*g on {displaystyle N.} Since {displaystyle pi } is a local diffeomorphism, Myers' theorem applies to the Riemannian manifold (N,π*g) and hence {displaystyle N} is compact. This implies that the fundamental group of {displaystyle M} is finite.

Cheng's diameter rigidity theorem The conclusion of Myers' theorem says that for any {displaystyle p,qin M,} one has dg(p,q) ≤ π/√k. In 1975, Shiu-Yuen Cheng proved: Let {displaystyle (M,g)} be a complete and smooth Riemannian manifold of dimension n. If k is a positive number with Ricg ≥ (n-1)k, and if there exists p and q in M with dg(p,q) = π/√k, then (M,g) is simply-connected and has constant sectional curvature k.

See also Gromov's compactness theorem (geometry) – On when a set of compact Riemannian manifolds of a given dimension is relatively compact References Ambrose, W. A theorem of Myers. Duke Math. J. 24 (1957), 345–348. Cheng, Shiu Yuen (1975), "Eigenvalue comparison theorems and its geometric applications", Mathematische Zeitschrift, 143 (3): 289–297, doi:10.1007/BF01214381, ISSN 0025-5874, MR 0378001 do Carmo, M. P. (1992), Riemannian Geometry, Boston, Mass.: Birkhäuser, ISBN 0-8176-3490-8 Myers, S. B. (1941), "Riemannian manifolds with positive mean curvature", Duke Mathematical Journal, 8 (2): 401–404, doi:10.1215/S0012-7094-41-00832-3 hide vte Manifolds (Glossary) Basic concepts Topological manifold AtlasDifferentiable/Smooth manifold Differential structureSmooth atlasSubmanifoldRiemannian manifoldSmooth mapSubmersionPushforwardTangent spaceDifferential formVector field Main results (list) Atiyah–Singer indexDarboux'sDe Rham'sFrobeniusGeneralized StokesHopf–RinowNoether'sSard'sWhitney embedding Maps CurveDiffeomorphism LocalGeodesicExponential map in Lie theoryFoliationImmersionIntegral curveLie derivativeSectionSubmersion Types of manifolds Closed(Almost) Complex(Almost) ContactFiberedFinslerFlatG-structureHadamardHermitianHyperbolicKählerKenmotsuLie group Lie algebraManifold with boundaryOrientedParallelizablePoissonPrimeQuaternionicHypercomplex(Pseudo−, Sub−) RiemannianRizza(Almost) SymplecticTame Tensors Vectors DistributionLie bracketPushforwardTangent space bundleTorsionVector fieldVector flow Covectors Closed/ExactCovariant derivativeCotangent space bundleDe Rham cohomologyDifferential form Vector-valuedExterior derivativeInterior productPullbackRicci curvature flowRiemann curvature tensorTensor field densityVolume formWedge product Bundles AdjointAffineAssociatedCotangentDualFiber(Co) FibrationJetLie algebra(Stable) NormalPrincipalSpinorSubbundleTangentTensorVector Connections AffineCartanEhresmannFormGeneralizedKoszulLevi-CivitaPrincipalVectorParallel transport Related Classification of manifoldsGauge theoryHistoryMorse theoryMoving frameSingularity theory Generalizations Banach manifoldDiffeologyDiffietyFréchet manifoldK-theoryOrbifoldSecondary calculus over commutative algebrasSheafStratifoldSupermanifoldTopologically stratified space Categories: Differential geometryGeometric inequalitiesTheorems in Riemannian geometry