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 $f:X\to Y$ 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 $df_{x}:T_{x}X\to T_{y}Y$ is surjective. Here, $T_{x}X$ and $T_{y}Y$ 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

^ Tu, Loring W. (2010), "9.3 The Regular Level Set Theorem", An Introduction to Manifolds, Springer, pp. 105–106, ISBN 9781441974006.
^ Banyaga, Augustin (2004), "Corollary 5.9 (The Preimage Theorem)", Lectures on Morse Homology, Texts in the Mathematical Sciences, 29, Springer, p. 130, ISBN 9781402026959.

[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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