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

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Boundary-defining functions exist locally and detect inward vectors

Statement

Every boundary point has a boundary-defining function; for any such $ρ$ , a vector $v$ is inward exactly when $d ρ_{p} (v) > 0$ .

Facts & Assumptions

Given: A boundary point $p$ of a smooth manifold $M$ , a boundary chart at $p$ , and, for the sign assertion, a boundary-defining function $ρ$ near $p$ and a vector $v \in T_{p} M$ .

[L1]

A boundary-defining function is nonnegative, vanishes exactly on the boundary, and has nonzero differential there (Boundary-defining functions).

[L2]

Inward, outward, and boundary-tangent vectors have respectively positive, negative, and zero last coordinate in a boundary chart (Inward, outward, and boundary-tangent vectors).

Proof

technique · direct

1.1givenL1construct

In the chosen boundary chart take $ρ = x^{n}$ . It is nonnegative, vanishes precisely on the face, and has nonzero differential, so it is a boundary-defining function by [L1].

2.1givenL1L2step 1.1∎

For any boundary-defining function $ρ$ , its restriction to the face is zero, so $d ρ_{p}$ vanishes on the tangent hyperplane. Its derivative in the positive normal coordinate is nonnegative because $ρ \geq 0$ on the half-space and $ρ (p) = 0$ ; by [L1] it is nonzero, hence positive. Therefore $d ρ_{p} (v)$ has the sign of the last coordinate of $v$ , and [L2] gives $d ρ_{p} (v) > 0$ exactly for inward $v$ .

Depends on

Used by

A circle is the boundary of a disk Example
The pair of pants is a cobordism realizing addition of circles Example

Dependency tree · two levels

10 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)