CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 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.

Transverse embedded submanifolds intersect in the expected codimension

Statement

If $S, T \subseteq M$ are transverse embedded submanifolds of codimensions $a$ and $b$ , then $S \cap T$ is an embedded submanifold of codimension $a + b$ . At each $p \in S \cap T$ ,

T_{p} (S \cap T) = T_{p} S \cap T_{p} T .

Facts & Assumptions

Given: Embedded submanifolds $S, T \subseteq M$ with $S ⋔ T$ .

[F1]

Two embedded submanifolds are transverse exactly when their inclusion maps are transverse (Transverse embedded submanifolds).

[L1]

The transverse preimage theorem identifies the preimage tangent space with the inverse image of the target tangent space (The transverse preimage theorem).

Proof

technique · direct

1.1

Let $i : S ↪ M$ be the inclusion. By [F1], $i ⋔ T$ . Since $i^{- 1} (T) = S \cap T$ , [L1] shows that $S \cap T$ is an embedded submanifold of $S$ of codimension ${codim}_{M} T = b$ .

F1L1given

2.1

Therefore ${codim}_{M} (S \cap T) = {codim}_{M} S + {codim}_{S} (S \cap T) = a + b .$ The tangent-space formula from [L1] becomes $T_{p} (S \cap T) = {v \in T_{p} S : d i_{p} (v) \in T_{p} T} = T_{p} S \cap T_{p} T .$

L1step 1.1algebra

3.1

Hence transverse embedded submanifolds intersect in the expected codimension.

step 2.1∎

Depends on

Used by

The intersection of coordinate spheres as a transverse level set Example
Transverse and tangent intersections of plane curves Example
Intersecting submanifolds need not be transverse False statement
A globally one-to-one transverse-fibre submanifold is a graph Proposition

Dependency tree · two levels

10 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

Marco Gualtieri, Topology I: Smooth Manifolds, cumulative notes (standard reference, not scraped)