DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-09-01

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A smooth map transverse to an embedded submanifold

Definition

Let $F : M \to N$ be smooth, and let $Z \subseteq N$ be an embedded submanifold. Then $F$ is transverse to $Z$ , written $F ⋔ Z$ , when for every $p \in F^{- 1} (Z)$ ,

$d F_{p} (T_{p} M) + T_{F (p)} Z = T_{F (p)} N .$

If $F (M) \cap Z = \emptyset$ , this condition is vacuous, so $F ⋔ Z$ still holds.

Depends on

Used by

A submersion is transverse to every embedded submanifold Corollary
A map vacuously transverse to a submanifold it avoids Example
Transversality is equivalent to surjectivity on the normal quotient Lemma
Transversality is invariant under diffeomorphic change of source and target Proposition
Transversality is stable on a compact source Proposition
Transversality to a point is the regular-value condition Proposition
Parametric transversality Theorem

Dependency tree · two levels

5 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)