Preimage theorem

In mathematics, particularly in 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 be a smooth map between manifolds. We say that a point is a regular value of f if for all the map is surjective. Here, and are the tangent spaces of X and Y at the points x and y.


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

References

  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, 29, Springer, p. 130, ISBN 9781402026959.
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