Preimage theorem

Preimage theorem In mathematics, particularly in the field of differential topology, the preimage theorem is a variation of the implicit function theorem concerning the preimage of particular points in a manifold under the action of a smooth map.[1][2] Statement of Theorem Definition. Permettere {stile di visualizzazione f:Xth Y} be a smooth map between manifolds. We say that a point {displaystyle yin Y} is a regular value of {stile di visualizzazione f} if for all {displaystyle xin f^{-1}(y)} the map {stile di visualizzazione df_{X}:T_{X}Xto T_{y}Y} is surjective. Qui, {stile di visualizzazione T_{X}X} e {stile di visualizzazione T_{y}Y} are the tangent spaces of {stile di visualizzazione X} e {stile di visualizzazione Y} at the points {stile di visualizzazione x} e {displaystyle y.} Teorema. Permettere {stile di visualizzazione f:Xth Y} be a smooth map, e lascia {displaystyle yin Y} be a regular value of {stile di visualizzazione f.} Quindi {stile di visualizzazione f^{-1}(y)} is a submanifold of {stile di visualizzazione X.} Se {displaystyle yin {testo{io sono}}(f),} then the codimension of {stile di visualizzazione f^{-1}(y)} is equal to the dimension of {stile di visualizzazione Y.} Anche, the tangent space of {stile di visualizzazione f^{-1}(y)} a {stile di visualizzazione x} è uguale a {displaystyle ker(df_{X}).} There is also a complex version of this theorem:[3] Teorema. Permettere {stile di visualizzazione X^{n}} e {stile di visualizzazione Y^{m}} be two complex manifolds of complex dimensions {displaystyle n>m.} Permettere {stile di visualizzazione g:Xth Y} be a holomorphic map and let {displaystyle yin {testo{io sono}}(g)} be such that {stile di visualizzazione {testo{rango}}(dg_{X})= m} per tutti {displaystyle xin g^{-1}(y).} Quindi {stile di visualizzazione g^{-1}(y)} is a complex submanifold of {stile di visualizzazione X} of complex dimension {displaystyle n-m.} See also Fiber (matematica) – Set of all points in a function's domain that all map to some single given point Level set – Subset of a function's domain on which its value is equal References ^ Tu, Loring W. (2010), "9.3 The Regular Level Set Theorem", An Introduction to Manifolds, Springer, pp. 105–106, ISBN 9781441974006. ^ Banyaga, Augustin (2004), "Corollario 5.9 (The Preimage Theorem)", Lectures on Morse Homology, Texts in the Mathematical Sciences, vol. 29, Springer, p. 130, ISBN 9781402026959. ^ Ferrari, Michele (2013), "Teorema 2.5", Varietà complesse - Lecture notes based on the course by Lambertus Van Geemen (PDF). nascondi i collettori (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 This topology-related article is a stub. Puoi aiutare Wikipedia espandendolo.

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