DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedaudited 2026-09-07

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Determinant-line orientations of finite-dimensional real vector spaces

Definition

For a finite-dimensional real vector space $V$ , an orientation is a positive ray in the one-dimensional determinant line $\det V = Λ^{\dim V} V$ . If $\dim V = 0$ , $\det V = R$ still has the two rays $R_{> 0}$ and $R_{< 0}$ .

Depends on

Used by

Chart integral with its orientation sign Definition
Induced boundary orientation Definition
Local orientation sign of a regular preimage Definition
Orientation induced on a hypersurface by a coorientation Definition
Oriented smooth manifolds and oriented charts Definition
Product orientations Definition
Standard orientation of the affine simplex Definition
The local oriented intersection sign Definition
The orientation line of a Morse critical point Definition
Positive-dimensional real projective space is orientable exactly in odd dimension Example
Orientation lines orient the continuation moduli spaces compatibly with gluing Lemma
Oriented boundary counts of a compact oriented 1-manifold cancel Lemma
Preimage orientation agrees with the local intersection sign Lemma
Swapping direct summands scales oriented bases by a sign Lemma
An oriented transverse normal bundle orients an embedded submanifold Proposition
Orientations and positive basis classes agree in positive dimension Proposition

Dependency tree · two levels

9 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)