Preimage theorem

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

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 ${\displaystyle f}$ 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 ${\displaystyle X}$ and ${\displaystyle Y}$ at the points ${\displaystyle x}$ and ${\displaystyle y}$.

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