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Preimage theorem

From Wikipedia, the free encyclopedia

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

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Definition. Let be a smooth map between manifolds. We say that a point is a regular value of if for all the map is surjective. Here, and are the tangent spaces of and at the points and

Theorem. Let be a smooth map, and let be a regular value of Then is a submanifold of If then the codimension of is equal to the dimension of Also, the tangent space of at is equal to

There is also a complex version of this theorem:[3]

Theorem. Let and be two complex manifolds of complex dimensions Let be a holomorphic map and let be such that for all Then is a complex submanifold of of complex dimension

See also

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  • 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

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  1. ^ Tu, Loring W. (2010), "9.3 The Regular Level Set Theorem", An Introduction to Manifolds, Springer, pp. 105–106, ISBN 9781441974006.
  2. ^ 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.
  3. ^ Ferrari, Michele (2013), "Theorem 2.5", Complex manifolds - Lecture notes based on the course by Lambertus Van Geemen (PDF).