Alphabeta MathHow 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.
A vector bundle is trivial if and only if it has a global frame
Statement
A smooth rank- vector bundle is trivial if and only if it has a global frame.
Facts & Assumptions
Given: A smooth rank- vector bundle .
Local frames and local trivializations are equivalent data on any open set (Local frames and local trivializations are equivalent data).
Proof
If is trivial, then the global bundle chart exists. Applying [L1] on to that chart produces a global frame.
If has a global frame, then applying [L1] on to that frame gives a global trivialization . Therefore is trivial.
Steps 1.1 and 1.2 prove both directions of the biconditional.
Depends on
Used by
- A closed manifold with formally plausible rank data needs positive codimension Counterexample
- Vanishing stable characteristic classes do not make two embeddings isotopic Counterexample
- An open parallelizable manifold immerses in Euclidean space of equal dimension Example
- Assuming countable choice, the normal bundle of the sphere is trivial Example
- Assuming countable choice, the tangent bundle of the circle is trivial Example
- Boy's surface: an immersion of the real projective plane in three-space Example
- Parallelizable tori have trivial stable normal class Example
- Pullback of the tautological line bundle along the antipodal cover Example
- The normal line of an oriented hypersurface is trivial Example
- Every vector bundle is globally trivial False statement
- Formal immersions of the circle in the plane are classified by the winding number Lemma
- Stably trivial bundles over spheres below the rank are trivial Lemma
- The framing obstruction lives in the normal bundle of the surgery sphere Lemma
- The pullback of a trivial smooth vector bundle is canonically trivial Lemma
- Euclidean formal immersions are homotopy equivalent to Stiefel-bundle sections Proposition
- Smale's classification of sphere immersions in Euclidean space Theorem
Dependency tree · two levels
8 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
- John M. Lee, Introduction to Smooth Manifolds (standard reference, not scraped)
- Will J. Merry, Differential Geometry (standard reference, not scraped)