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

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Boundary orientation is independent of the outward vector field

Statement

The outward-normal-first boundary orientation is independent of the chosen outward vector field. On the interval $[a, b]$ with its standard orientation, it gives $\partial [a, b] = {b} - {a}$ .

Facts & Assumptions

Given: An oriented smooth manifold $M$ with boundary and two outward vectors $n_{0}, n_{1} \in T_{p} M$ at a boundary point $p$ ; for the final assertion, the standard orientation on $[a, b]$ .

[L1]

Boundary orientation is defined by the outward-normal-first rule (Induced boundary orientation).

[L2]

The boundary-tangent vectors form a hyperplane in $T_{p} M$ , and outward vectors lie in the same negative normal half-space (The boundary tangent space is the boundary-tangent hyperplane; Inward, outward, and boundary-tangent vectors).

Proof

technique · direct

1.1givenL2algebra

By [L2], the images of $n_{0}$ and $n_{1}$ in the one-dimensional quotient $T_{p} M / T_{p} \partial M$ lie on the same ray. Hence $n_{1} = a n_{0} + w$ for some $a > 0$ and $w \in T_{p} \partial M$ .

2.1givenL1step 1.1algebra∎

If $τ$ is a nonzero boundary determinant, alternation gives $n_{1} \land τ = a (n_{0} \land τ)$ because $w \land τ = 0$ . Thus [L1] is independent of the outward vector. At $b$ the outward vector is $+ \partial_{x}$ , while at $a$ it is $- \partial_{x}$ , so [L1] gives $\partial [a, b] = {b} - {a}$ .

Depends on

Used by

Oriented smooth cobordism Definition
Boundary orientation of the unit sphere by the outward normal Example
Signed points give the oriented zero-bordism invariant Example
The boundary of an oriented interval Example
Boundary orientation is inward-normal-first False statement
Boundary orientation of the compactified one-dimensional Morse moduli space Lemma
Collar gluing and seam smoothing give transitivity Lemma
Cylinders give reflexivity of cobordism Lemma
Oriented boundary counts of a compact oriented 1-manifold cancel Lemma
Oriented boundary of an intersection trace has opposite end signs Lemma
Reversing a cobordism gives symmetry Lemma
A leafwise positive closed two-form calibrates a taut foliation Proposition
Boundary orientation of a product with at most one boundary factor Proposition
Zero-dimensional bordism groups Proposition

Dependency tree · two levels

7 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)