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 sections, local sections, and support

Definition

Let $π : E \to M$ be a smooth vector bundle.

A smooth section of $E$ is a smooth map $s : M \to E$ such that $π \circ s = {id}_{M}$ .

If $U \subseteq M$ is open, a smooth local section on $U$ is a smooth map $s : U \to E ∣_{U}$ with $π \circ s = {id}_{U}$ .

The support of a section $s$ is the closure of the set ${p \in M : s (p) \neq 0_{p}}$ . A section is compactly supported when this set has compact closure.

Depends on

Used by

Every smooth vector field on a compact manifold is complete Corollary
A nonproper pullback destroys compact support Counterexample
A smooth differential k-form Definition
A smooth tensor field Definition
A smooth vector field is a smooth section of the tangent bundle Definition
Cech cohomology of holomorphic sections of a line bundle on finite good covers Definition
Complex manifolds, holomorphic vector bundles and Hermitian metrics Definition
Connection on a smooth vector bundle Definition
Holomorphic line bundles and meromorphic sections on a Riemann surface Definition
Interior product of a form by a vector field Definition
Local and global frames of a vector bundle Definition
Second-order Laplace-type operators on metric vector bundles Definition
The Bott partial connection on the normal bundle of a foliation Definition
The maximal Dolbeault operator and its Hilbert adjoint on a compact Riemann surface Definition
The self-intersection number of a complementary-dimensional oriented submanifold Definition
Self-intersection of the zero section in an oriented plane bundle Example
The diagonal in the two-sphere has self-intersection two Example
The trivial line bundle and its sections as functions Example
Pullback of a compactly supported form is always compactly supported False statement
Compactness gives a compactly supported time-dependent velocity field Lemma
Formal adjoints of differential operators on metric vector bundles Lemma
Pullback of the Thom class along a transverse section computes the Euler class Lemma
Euclidean formal immersions are homotopy equivalent to Stiefel-bundle sections Proposition
Smooth sections form a module over smooth functions Proposition
Smoothness of a section is equivalent to smooth local components Proposition
The exterior derivative does not enlarge support Proposition
Compactly supported smooth vector fields are complete Theorem
Compactly supported time-dependent vector fields have global evolution on a compact time interval Theorem
Harmonic representatives for compact de Rham and Dolbeault complexes Theorem

Dependency tree · two levels

7 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)