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.

Smooth vector bundles, rank, fibres, and trivial bundles

Definition

Let $π : E \to M$ be a smooth fibre bundle and let $r \in N$ .

A smooth vector bundle of rank $r$ is a smooth fibre bundle for which every fibre $E_{p} : = π^{- 1} (p)$ is an $r$ -dimensional real vector space and there is an open cover $(U_{α})$ of $M$ such that each restriction $π^{- 1} (U_{α}) \to U_{α}$ admits a local trivialization

$Φ_{α} : π^{- 1} (U_{α}) \to U_{α} \times R^{r}$

whose restriction on each fibre $E_{p}$ is a linear isomorphism $E_{p} \to R^{r}$ .

The fibre over $p \in M$ is called the fibre at $p$ . A vector bundle is trivial when it is globally isomorphic over $M$ to the product bundle $M \times R^{r} \to M$ .

Depends on

Used by

A bundle map with rank drop is not a formal immersion Counterexample
A closed manifold with formally plausible rank data needs positive codimension Counterexample
Connections can change a representative without changing its class Counterexample
The Mobius core circle has no integral oriented self-intersection but mod two data survives Counterexample
Complex manifolds, holomorphic vector bundles and Hermitian metrics Definition
Complex-linear and metric-compatible bundle connections Definition
Connection on a smooth vector bundle Definition
Dual and Hom vector bundles Definition
Formal immersion between smooth manifolds Definition
Framed embedded surgery sphere Definition
Framings of a normal bundle Definition
Holomorphic line bundles and meromorphic sections on a Riemann surface Definition
Pullback vector bundles as fibre products Definition
Restrictions of vector bundles Definition
Second-order Laplace-type operators on metric vector bundles Definition
Smooth bundle metrics Definition
Smooth sections, local sections, and support Definition
Stable normal inverse of the tangent bundle Definition
The holomorphic line bundle associated to a divisor Definition
Vector bundle charts and transition functions Definition
Vector bundle maps over a smooth base map Definition
Vector subbundles Definition
Whitney sums of vector bundles Definition
An open parallelizable manifold immerses in Euclidean space of equal dimension Example
Assuming countable choice, the tangent and cotangent bundles are smooth vector bundles Example
Curvature and first Chern form of a complex line Example
Flat connections and real characteristic classes Example
Self-intersection of the zero section in an oriented plane bundle Example
The standard sphere immersion and its normal line Example
The trivial line bundle and its sections as functions Example
An embedding into Euclidean space gives a rank-(n-m) stable normal inverse Lemma
Complex flag splitting with injective real pullback on smooth bases Lemma
Disk bundles over compact bases are compact manifolds with boundary Lemma
First Chern form agrees with the topological line class Lemma
Formal adjoints of differential operators on metric vector bundles Lemma
Normal bundle of the zero locus of a transverse section Lemma
Oriented real two-plane splitting with real-cohomology injection Lemma
Smooth manifolds have CW homotopy type Lemma
Stably trivial bundles over spheres below the rank are trivial Lemma
The boundary stable tangent bundle splits off a trivial line Lemma

…and 15 more results.

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