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

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The boundary tangent space is the boundary-tangent hyperplane

Statement

For $p \in \partial M$ of an $n \geq 1$ dimensional manifold and the inclusion $i : \partial M ↪ M$ , the differential $d i_{p}$ identifies $T_{p} \partial M$ with the hyperplane of boundary-tangent vectors in $T_{p} M$ .

Facts & Assumptions

Given: An $n$ -dimensional smooth manifold $M$ with boundary, where $n \geq 1$ , and a point $p \in \partial M$ .

[L1]

The boundary is an embedded smooth $(n - 1)$ -manifold with charts obtained by restricting boundary charts to their faces (The boundary of a positive-dimensional manifold is a closed embedded smooth (n-1)-manifold).

[L2]

The full tangent space $T_{p} M$ has the $n$ boundary-chart coordinate derivations as a basis (Tangent and cotangent bundles extend over a boundary).

[L3]

Boundary-tangent vectors are exactly those with zero last coordinate in a boundary chart (Inward, outward, and boundary-tangent vectors).

Proof

technique · direct

1.1givenL1constructalgebra

By [L1], a restricted face chart on $\partial M$ has coordinate vectors $\partial_{1}, \dots, \partial_{n - 1}$ . In the corresponding boundary chart on $M$ , the inclusion is $i (x^{1}, \dots, x^{n - 1}) = (x^{1}, \dots, x^{n - 1}, 0)$ , so $d i_{p}$ sends those vectors to the first $n - 1$ ambient coordinate derivations. Hence $d i_{p}$ is injective and its image is their span.

2.1givenL2L3step 1.1∎

In the full basis from [L2], the image found in step 1.1 is precisely the last-coordinate-zero hyperplane, which [L3] identifies with the boundary-tangent vectors.

Depends on

Used by

Induced boundary orientation Definition
The tangent space at a boundary point has dimension n-1 False statement
Restriction of a foliation transverse to the boundary Lemma
The boundary stable tangent bundle splits off a trivial line Lemma
Boundary orientation is independent of the outward vector field Proposition
Boundary-tangent fields have boundary-preserving local two-sided flows Theorem
Riemannian divergence 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)