Preimage theorem

In mathematics, particularly in the field of differential topology, the preimage theorem, or regular value theorem, is a variation of the implicit function theorem concerning the preimage of particular points in a smooth manifold under the action of a smooth map.

Statement of Theorem

Definition. Let ${\displaystyle X}$ and ${\displaystyle Y}$ be smooth manifolds, and let ${\displaystyle f:X\to Y}$ be smooth. 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 differential ${\displaystyle df_{x}:T_{x}X\to T_{y}Y}$ 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 ${\displaystyle y.}$ A submanifold ${\displaystyle S\subseteq X}$ is called smoothly embedded if the inclusion map ${\displaystyle \iota :S\hookrightarrow X}$ is a smooth embedding, and properly embedded if ${\displaystyle \iota }$ is a proper map.

Theorem. Let ${\displaystyle X}$ and ${\displaystyle Y}$ be smooth manifolds, and let ${\displaystyle f:X\to Y}$ be smooth. If ${\displaystyle y\in Y}$ is a regular value of ${\displaystyle f}$, then ${\displaystyle S=f^{-1}(y)}$ is a smoothly and properly embedded submanifold of ${\displaystyle X.}$ In addition, the codimension of ${\displaystyle S}$ is equal to the dimension of ${\displaystyle Y}$, and if ${\displaystyle x\in S}$, then ${\displaystyle T_{x}S=\ker(df_{x})}$, where ${\displaystyle \ker(df_{x})}$ is the kernel of the differential of ${\displaystyle f}$ at ${\displaystyle x}$.

There is also a complex version of this theorem:

Theorem. Let ${\displaystyle X^{n}}$ and ${\displaystyle Y^{m}}$ be two complex manifolds of complex dimensions ${\displaystyle n>m.}$ Let ${\displaystyle g:X\to Y}$ be a holomorphic map and let ${\displaystyle y\in {\text{im}}(g)}$ be such that ${\displaystyle {\text{rank}}(dg_{x})=m}$ for all ${\displaystyle x\in g^{-1}(y).}$ Then ${\displaystyle g^{-1}(y)}$ is a complex submanifold of ${\displaystyle X}$ of complex dimension ${\displaystyle n-m.}$

See also

• Fiber (mathematics) – Class of points in a function's domain defined by having the same image by that map
• Level set – Subset of a function's domain on which its value is equal

References

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