TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-09-01

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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The transverse preimage theorem

Statement

Let $F : M^{m} \to N^{n}$ be smooth and let $Z \subseteq N$ be an embedded submanifold of codimension $c$ . If $F ⋔ Z$ , then $F^{- 1} (Z)$ is an embedded submanifold of $M$ of codimension $c$ . For each $p \in F^{- 1} (Z)$ ,

T_{p} F^{- 1} (Z) = {v \in T_{p} M : d F_{p} (v) \in T_{F (p)} Z} .

Facts & Assumptions

Given: A smooth map $F : M^{m} \to N^{n}$ transverse to an embedded submanifold $Z \subseteq N$ .

[L1]

Embedded submanifolds admit local defining submersions (Embedded submanifolds admit local defining submersions).

[L2]

Transversality to $Z$ is equivalent to surjectivity on the normal quotient (Transversality is equivalent to surjectivity on the normal quotient).

[L3]

A regular level set is an embedded submanifold, and its tangent space is the kernel of the defining submersion differential (A regular level set is an embedded submanifold, The tangent space of a regular level set is the kernel).

Proof

technique · direct

1.1

Fix $p \in F^{- 1} (Z)$ and put $y = F (p)$ . By [L1], choose a neighbourhood $V$ of $y$ and a submersion $h : V \to R^{c}$ such that $Z \cap V = h^{- 1} (0)$ .

L1givenchoose

2.1

At $p$ , the differential $d h_{y}$ kills exactly $T_{y} Z$ . Therefore [L2] implies that $d h_{y} \circ d F_{p} = d (h \circ F)_{p}$ is surjective. So $0$ is a regular value of $h \circ F$ near $p$ .

L2step 1.1algebra

3.1

Since $(h \circ F)^{- 1} (0) = F^{- 1} (Z) \cap F^{- 1} (V)$ , [L3] shows that this set is an embedded codimension- $c$ submanifold near $p$ , with tangent space $\ker d (h \circ F)_{p} = {v : d F_{p} (v) \in T_{y} Z} .$

L3step 2.1algebra

4.1

Because $p$ was arbitrary, these local models glue to an embedded submanifold structure on $F^{- 1} (Z)$ with the stated tangent-space formula.

step 3.1∎

Depends on

Used by

Transverse embedded submanifolds intersect in the expected codimension Corollary
The intersection of coordinate spheres as a transverse level set Example
A preimage need not be a submanifold without transversality False statement
Parametric transversality Theorem
Transverse fibre products are embedded submanifolds Theorem

Dependency tree · two levels

13 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John M. Lee, Introduction to Smooth Manifolds, 2nd ed. (standard reference, not scraped)
Marco Gualtieri, Topology I: Smooth Manifolds, cumulative notes (standard reference, not scraped)