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

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Boundary orientation of a product with at most one boundary factor

Statement

For oriented $M^{m}, N$ , if $\partial N = \emptyset$ then $\partial (M \times N) = \partial M \times N$ has the product boundary orientation; if $\partial M = \emptyset$ then $M \times \partial N$ has $(- 1)^{m}$ times the product orientation.

Facts & Assumptions

Given: Oriented manifolds $M^{m}$ and $N$ , with at most one of $\partial M$ and $\partial N$ nonempty.

[L1]

The product orientation uses the ordered determinant $\det (T M \oplus T N) ≅ \det (T M) \otimes \det (T N)$ (Product orientations).

[L2]

Boundary orientation places an outward normal before a positive boundary determinant (Induced boundary orientation).

Proof

technique · direct

1.1

By [L1] and [L2], compare the boundary orientation with the product orientation by moving the outward normal of the boundary factor to the first position in the ordered determinant of $T M \oplus T N$ .

givenL1L2

2.1

On $\partial M \times N$ , the normal is already first, so the two orientations agree. On $M \times \partial N$ , it crosses the $m$ tangent vectors from $M$ , producing $(- 1)^{m}$ . If both boundaries are nonempty, their product has corners and lies outside the stated hypotheses.

step 1.1algebra∎

Depends on

Used by

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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)