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

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A global inward-pointing boundary vector field exists

Statement

Assume ${A C}_{ω}$ . Let $M$ be a smooth manifold with boundary. There is a smooth vector field, defined on a neighbourhood of $\partial M$ , which is inward at every boundary point.

Facts & Assumptions

Given: ${A C}_{ω}$ and a smooth manifold $M$ with boundary.

Proof

technique · direct

1.1

Choose boundary-chart neighbourhoods covering $\partial M$ and, on each one, take the coordinate field with positive last component. Add $Int M$ to this cover and choose a smooth partition of unity subordinate to it.

given

2.1

Extend each partition-weighted coordinate field by zero outside its chart and sum the locally finite family on a neighbourhood of $\partial M$ ; the term supported in $Int M$ vanishes there. At a boundary point its normal component is a positive weighted sum of positive numbers, hence is positive. Thus this directly constructed smooth field is inward everywhere on the boundary.

step 1.1∎

Depends on

Used by

Collar neighborhood theorem Theorem

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

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)