Alphabeta MathDefinitionDefinition: 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.
Orientable manifolds
Definition
A smooth manifold is orientable if it admits an orientation. This asserts existence, not a preferred orientation and not a single hidden choice across components.
Depends on
Used by
- A reflected sphere embedding is regularly homotopic but not isotopic to the standard one Counterexample
- Vanishing stable characteristic classes do not make two embeddings isotopic Counterexample
- Transversely oriented codimension-one foliations Definition
- Boy's surface: an immersion of the real projective plane in three-space Example
- Positive-dimensional real projective space is orientable exactly in odd dimension Example
- The Mobius band is nonorientable although its boundary circle is orientable Example
- The normal line of an oriented hypersurface is trivial Example
- Every manifold is orientable False statement
- An ambient isotopy preserves the orientation of an invariant round sphere Lemma
- The components of the frame bundle of a connected manifold Lemma
- The orientation double cover is canonically oriented and preserves closedness Lemma
- Nonempty connected orientable manifolds have exactly two orientations Proposition
- Integral Morse homology does not require orientability of the manifold Remark
- Sphere eversion cannot be an isotopy through embeddings Remark
- Framed zero-dimensional bordism in a nonorientable manifold is mod two Theorem
- Orientability is equivalent to a nowhere-vanishing top form Theorem
- The Hopf mod-two degree theorem for nonorientable domains Theorem
Dependency tree · two levels
4 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)