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. Let {displaystyle f:Xto Y} be a smooth map between manifolds. We say that a point {displaystyle yin Y} is a regular value of {displaystyle f} if for all {displaystyle xin f^{-1}(y)} the map {displaystyle df_{x}:T_{x}Xto T_{y}Y} is surjective. Here, {displaystyle T_{x}X} and {displaystyle T_{y}Y} are the tangent spaces of {displaystyle X} and {displaystyle Y} at the points {displaystyle x} and {displaystyle y.} Theorem. Let {displaystyle f:Xto Y} be a smooth map, and let {displaystyle yin Y} be a regular value of {displaystyle f.} Then {displaystyle f^{-1}(y)} is a submanifold of {displaystyle X.} If {displaystyle yin {text{im}}(f),} then the codimension of {displaystyle f^{-1}(y)} is equal to the dimension of {displaystyle Y.} Also, the tangent space of {displaystyle f^{-1}(y)} at {displaystyle x} is equal to {displaystyle ker(df_{x}).} There is also a complex version of this theorem:[3] Theorem. Let {displaystyle X^{n}} and {displaystyle Y^{m}} be two complex manifolds of complex dimensions {displaystyle n>m.} Let {displaystyle g:Xto Y} be a holomorphic map and let {displaystyle yin {text{im}}(g)} be such that {displaystyle {text{rank}}(dg_{x})=m} for all {displaystyle xin g^{-1}(y).} Then {displaystyle g^{-1}(y)} is a complex submanifold of {displaystyle X} of complex dimension {displaystyle n-m.} See also Fiber (mathematics) – 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), "Corollary 5.9 (The Preimage Theorem)", Lectures on Morse Homology, Texts in the Mathematical Sciences, vol. 29, Springer, p. 130, ISBN 9781402026959. ^ Ferrari, Michele (2013), "Theorem 2.5", Complex manifolds - Lecture notes based on the course by Lambertus Van Geemen (PDF). 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 This topology-related article is a stub. You can help Wikipedia by expanding it.

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