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

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Half-space extensions agreeing on a relatively open set have the same derivatives there

Statement

If two smooth Euclidean extensions agree on a relatively open subset of $H^{n}$ , then all of their derivatives agree at every point of that subset.

Facts & Assumptions

Given: A relatively open set $U \subseteq H^{n}$ and two smooth Euclidean maps $F$ and $G$ , defined on neighbourhoods of $U$ , whose restrictions to $U$ agree.

Proof

technique · direct

1.1

Their difference vanishes on the relative interior, which is dense in the relative set. Every Euclidean derivative of the difference vanishes there by ordinary differentiation.

given

2.1

Those derivatives are continuous, so they also vanish at each face point. Hence the derivative of a half-space-smooth map is independent of its chosen extension.

step 1.1∎

Depends on

Smooth functions on relatively open half-space sets

Used by

Boundary smoothness is independent of charts and extensions Lemma
Chain rule for smooth half-space maps Proposition
Tangent and cotangent bundles extend over a boundary Theorem

Dependency tree · two levels

2 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)