LemmaStatement: AI-adaptedProof: AI-generatedprecheck passaudited 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.

Every vector in a fibre extends to a compactly supported smooth section

Statement

Let $π : E \to M$ be a smooth vector bundle, let $p \in M$ , and let $v \in E_{p}$ . Then there is a compactly supported smooth section $σ$ of $E$ with $σ (p) = v$ .

Facts & Assumptions

Given: A smooth vector bundle $E \to M$ , a point $p \in M$ , and a vector $v \in E_{p}$ .

[L1]

Around $p$ there is a local frame of $E$ (Local and global frames of a vector bundle).

[L2]

There is a smooth bump function equal to $1$ at $p$ and supported in a prescribed chart neighborhood (A chart bump at a point with prescribed support).

[L3]

Multiplying a smooth section by a smooth function keeps it smooth (Smooth sections form a module over smooth functions).

[L4]

A section is smooth exactly when its local frame components are smooth (Smoothness of a section is equivalent to smooth local components).

Proof

technique · direct

1.1L1givenchoose

Choose a local frame $(s_{1}, \dots, s_{r})$ on an open set $U$ containing $p$ . Write $v = \sum_{i} a_{i} s_{i} (p)$ and define a local section $τ = \sum_{i} a_{i} s_{i}$ on $U$ . Then $τ (p) = v$ .

2.1L2L3L4step 1.1construct∎

Choose a smooth bump function $χ : M \to R$ with $χ (p) = 1$ and $supp (χ) \subseteq U$ . On $U$ define $σ = χ τ$ , which is smooth by [L3]. Because $supp (χ) \subseteq U$ , there is an open neighborhood $V$ of $M ∖ U$ on which $χ = 0$ ; define $σ = 0$ on $V$ , which is smooth by [L4]. On $U \cap V$ the two formulas agree, so they paste to a smooth global section. Its support is contained in $supp (χ)$ , hence compact, and $σ (p) = χ (p) τ (p) = v$ .

Depends on

Used by

Dependency tree · two levels

18 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)