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

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

## 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.