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. Lassen {Anzeigestil f:X. Y} be a smooth map between manifolds. We say that a point {displaystyle yin Y} is a regular value of {Anzeigestil f} if for all {displaystyle xin f^{-1}(j)} the map {Anzeigestil df_{x}:T_{x}Xto T_{j}Y} is surjective. Hier, {Anzeigestil T_{x}X} und {Anzeigestil T_{j}Y} are the tangent spaces of {Anzeigestil X} und {Anzeigestil Y} at the points {Anzeigestil x} und {displaystyle y.} Satz. Lassen {Anzeigestil f:X. Y} be a smooth map, und lass {displaystyle yin Y} be a regular value of {displaystyle f.} Dann {Anzeigestil f^{-1}(j)} is a submanifold of {displaystyle X.} Wenn {displaystyle yin {Text{ich bin}}(f),} then the codimension of {Anzeigestil f^{-1}(j)} is equal to the dimension of {Anzeigestil Y.} Ebenfalls, the tangent space of {Anzeigestil f^{-1}(j)} bei {Anzeigestil x} ist gleich {displaystyle ker(df_{x}).} There is also a complex version of this theorem:[3] Satz. Lassen {Anzeigestil X^{n}} und {Anzeigestil Y^{m}} be two complex manifolds of complex dimensions {displaystyle n>m.} Lassen {Anzeigestil g:X. Y} be a holomorphic map and let {displaystyle yin {Text{ich bin}}(g)} be such that {Anzeigestil {Text{Rang}}(dg_{x})=m} für alle {displaystyle xin g^{-1}(j).} Dann {Anzeigestil g^{-1}(j)} is a complex submanifold of {Anzeigestil X} of complex dimension {displaystyle n-m.} See also Fiber (Mathematik) – 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), "Logische Folge 5.9 (The Preimage Theorem)", Lectures on Morse Homology, Texts in the Mathematical Sciences, vol. 29, Springer, p. 130, ISBN 9781402026959. ^ Ferrari, Michèle (2013), "Satz 2.5", Complex manifolds - Lecture notes based on the course by Lambertus Van Geemen (Pdf). vte Verteiler ausblenden (Glossar) Grundkonzepte Topologische Mannigfaltigkeit AtlasDifferentiable/Glatte Mannigfaltigkeit DifferentialstrukturGlatter AtlasUntermannigfaltigkeitRiemannsche MannigfaltigkeitGlatte KarteSubmersionPushforwardTangentenraumDifferentialformVektorfeld Hauptergebnisse (aufführen) Atiyah–Singer indexDarboux’sDe Rham’sFrobeniusGeneralized StokesHopf–RinowNoether’sSard’sWhitney embedding Maps CurveDiffeomorphism LocalGeodesicExponential map in Lie theoryFoliationImmersionIntegralkurveLie-AbleitungSectionSubmersion Types of manifolds Closed(Fast) Komplex(Fast) KontaktFiberedFinslerFlatG-StrukturHadamardHermitianHyperbolicKählerKenmotsuLie-Gruppe Lie-AlgebraManifold with borderOrientedParallelizablePoissonPrimeQuaternionicHypercomplex(Pseudo−, Unter−) RiemannianRizza(Fast) SymplecticTame Tensors Vectors DistributionLie BracketPushforwardTangential space bundleTorsionVector fieldVector flow Covectors Closed/ExactCovariant ableiteCotangent space bundleDe Rham cohomologyDifferential form Vector-valuedExterior derivativeInterior productPullbackRicci Curvature flowRiemann curvature tensorTensor field densityVolume formWedge product Bundles AdjointAffineAssociatedCotangentDualFiber(Co) FibrationJetLie-Algebra(Stabil) 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. Sie können Wikipedia helfen, indem Sie es erweitern.

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