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.

Oriented smooth manifolds and oriented charts

Definition

An orientation of an $n$ -manifold is a smooth choice of a ray in $\det T_{p} M$ for every $p$ . For $n > 0$ , a chart is oriented when its coordinate frame is in that ray. For $n = 0$ , the datum is a sign at each point; the unique empty frame does not encode both choices.

Depends on

Used by

Orientable manifolds Definition
Orientation-preserving parametrizations Definition
The Mobius band is nonorientable although its boundary circle is orientable Example
The product orientation on a torus Example
Every manifold is orientable False statement
Nonempty connected orientable manifolds have exactly two orientations Proposition
Pointwise orientation sign of a local diffeomorphism Proposition
Orientability is equivalent to a nowhere-vanishing top form Theorem
Positive oriented atlases characterize orientations except for one-manifolds with boundary 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)