CorollaryStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passaudited 2026-09-07

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 functions and tensor fields extend locally across the boundary

Statement

Every smooth function or tensor field on a manifold with boundary extends smoothly across each boundary point to some neighbourhood in its double; the extension is not canonical.

Facts & Assumptions

Given: A smooth manifold $M$ with boundary, a smooth function or smooth tensor field $T$ on $M$ , and a boundary point $p \in \partial M$ .

[L1]

A seam point of the smooth double has a chart identifying the two labelled halves with the two closed Euclidean half-spaces (The double has a well-defined smooth structure).

[L2]

A smooth map on a relatively open half-space set has a smooth Euclidean extension near each point (Smooth functions on relatively open half-space sets).

[L3]

A smooth tensor field is a smooth section of its tensor bundle (A smooth tensor field).

Proof

technique · direct

1.1

By [L1], choose a seam chart at $p$ in which the labelled copy of $M$ is a half-space. A function extends there by [L2]. For a tensor field, [L3] expresses it in the smooth coordinate frame with finitely many smooth component functions, each of which extends by [L2].

givenL1L2L3

2.1

Reassemble the extended components in the same coordinate frame and restrict to a smaller neighbourhood of $p$ in the double. The resulting tensor is smooth and restricts to $T$ on $M$ . Since [L2] supplies no unique Euclidean extension, this construction is not canonical.

L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

16 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

Ioan Mărcuț, Manifolds (2017 lecture notes), §§14.5, 15.1 (standard reference, not scraped)
Will Merry, Differential Geometry (2021), Lecture 24 (standard reference, not scraped)