Preimage theorem

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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[edit]

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 .


  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 .