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: Permettere {stile di visualizzazione (M,g)} be a complete Riemannian manifold of dimension {stile di visualizzazione 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 {stile di visualizzazione ,leq pi /{mq {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.

Contenuti 1 Corollari 2 Cheng's diameter rigidity theorem 3 Guarda anche 4 References Corollaries The conclusion of the theorem says, in particolare, that the diameter of {stile di visualizzazione (M,g)} è finito. The Hopf-Rinow theorem therefore implies that {stile di visualizzazione M} must be compact, as a closed (and hence compact) ball of radius {stile di visualizzazione pi /{mq {K}}} in any tangent space is carried onto all of {stile di visualizzazione 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 {stile di visualizzazione pi :Nto M.} One may consider the Riemannian metric π*g on {displaystyle N.} Da {stile di visualizzazione pi } is a local diffeomorphism, Myers' theorem applies to the Riemannian manifold (N,π*g) e quindi {stile di visualizzazione N} è compatto. This implies that the fundamental group of {stile di visualizzazione M} è finito.

Cheng's diameter rigidity theorem The conclusion of Myers' theorem says that for any {stile di visualizzazione p,qin M,} one has dg(p,q) ≤ π/√k. In 1975, Shiu-Yuen Cheng proved: Permettere {stile di visualizzazione (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, poi (M,g) is simply-connected and has constant sectional curvature k.

See also Gromov's compactness theorem (geometria) – On when a set of compact Riemannian manifolds of a given dimension is relatively compact References Ambrose, w. A theorem of Myers. Duca Matematica. J. 24 (1957), 345–348. Cheng, Shiu Yuen (1975), "Eigenvalue comparison theorems and its geometric applications", Giornale di matematica, 143 (3): 289–297, doi:10.1007/BF01214381, ISSN 0025-5874, SIG 0378001 fai Carmo, M. P. (1992), Riemannian Geometry, Boston, Messa.: Birkhauser, ISBN 0-8176-3490-8 Myers, S. B. (1941), "Riemannian manifolds with positive mean curvature", Diario di matematica del duca, 8 (2): 401–404, doi:10.1215/S0012-7094-41-00832-3 hide vte Manifolds (Glossario) Basic concepts Topological manifold AtlasDifferentiable/Smooth manifold Differential structureSmooth atlasSubmanifoldRiemannian manifoldSmooth mapSubmersionPushforwardTangent spaceDifferential formVector field Main results (elenco) Atiyah–Singer indexDarboux'sDe Rham'sFrobeniusGeneralized StokesHopf–RinowNoether'sSard'sWhitney embedding Maps CurveDiffeomorphism LocalGeodesicExponential map in Lie theoryFoliationImmersionIntegral curveLie derivativeSectionSubmersion Types of manifolds Closed(Quasi) Complex(Quasi) ContactFiberedFinslerFlatG-structureHadamardHermitianHyperbolicKählerKenmotsuLie group Lie algebraManifold with boundaryOrientedParallelizablePoissonPrimeQuaternionicHypercomplex(Pseudo-, Sub−) RiemannianRizza(Quasi) 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(Stabile) 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