DefinitionDefinition: AI-adaptedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-31

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.

Vector bundle maps over a smooth base map

Definition

Let $π_{E} : E \to M$ and $π_{F} : F \to N$ be smooth vector bundles, and let $f : M \to N$ be a smooth map.

A vector bundle map over $f$ is a smooth map $Φ : E \to F$ such that $π_{F} \circ Φ = f \circ π_{E}$ and, for every $p \in M$ , the restriction $Φ ∣_{E_{p}} : E_{p} \to F_{f (p)}$ is linear.

When $f = {id}_{M}$ , one also says that $Φ$ is a bundle map over the identity.

Depends on

Used by

A bundle map with rank drop is not a formal immersion Counterexample
Equivariant maps and equivariant vector bundles Definition
Formal immersion between smooth manifolds Definition
Gauss frame map of an immersion into Euclidean space Definition
Left Maurer--Cartan form Definition
Normal bundle of a formal immersion Definition
The derivative map from immersions to formal immersions Definition
Immersing the circle in the plane from a formal line monomorphism Example
For compact sources the immersion condition is open in the weak smooth topology Lemma
Normal bundle of the zero locus of a transverse section Lemma
Smoothing continuous families of formal immersions Lemma
A fibrewise bijective smooth bundle map over a diffeomorphism is a bundle isomorphism Proposition
Euclidean formal immersions are homotopy equivalent to Stiefel-bundle sections Proposition
Pullback is functorial up to canonical bundle isomorphism Proposition
Sections of Hom are the same as smooth fibrewise linear maps Proposition
Smoothness of a bundle map is equivalent to smooth local matrices Proposition
The canonical map to a quotient bundle is a smooth bundle map Proposition
The Euler class of an oriented even-rank normal bundle controls self-intersection Proposition

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 (standard reference, not scraped)
Will J. Merry, Differential Geometry (standard reference, not scraped)