Alphabeta Math
PropositionStatement: 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.
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Transversality to a point is the regular-value condition

Statement

For a smooth map F:M→N and a point q∈N, the condition F⋔{q} is equivalent to saying that q is a regular value of F.

Facts & Assumptions

Given: A smooth map F:M→N and a point q∈N.

[F1]

Transversality to {q} means dFp(TpM)+Tq{q}=TqN for every p∈F−1(q) (A smooth map transverse to an embedded submanifold).

[F2]

A regular value is one whose fibre points are all regular points, and regular points are exactly the submersion points (Regular and critical points and values).

Proof

technique · direct
1.1

Because Tq{q}=0, [F1] says that F⋔{q} exactly when dFp(TpM)=TqN for every p∈F−1(q).

F1given
2.1

The condition in step 1.1 is precisely that every fibre point is a submersion point, which [F2] identifies with q being a regular value.

F2step 1.1
3.1

Therefore transversality to a point is the regular-value condition.

step 2.1∎

Depends on

Used by

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