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 f: X \to Y\,\! be a smooth map between manifolds. We say that a point y \in Y is a regular value of f if for all x \in f^{-1}(y) the map df_x: T_xX \to T_yY\,\! is surjective. Here, T_xX\,\! and T_yY\,\! are the tangent spaces of X and Y at the points x and y.


Theorem. Let f: X \to Y\,\! be a smooth map, and let y \in Y be a regular value of f; then f^{-1}(y) is a submanifold of X. If y \in \text{im}(f), then the codimension of f^{-1}(y) is equal to the dimension of Y. Also, the tangent space of f^{-1}(y) at x is equal to \ker(df_x).

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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