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.
Cotangent transitions are inverse transpose, not tangent Jacobians
Statement
False claim: the cotangent bundle uses the same transition matrices as the tangent bundle.
Facts & Assumptions
Given: On , the overlapping global charts and .
Cotangent coordinate changes use the inverse transpose Jacobian (Cotangent coordinate changes use the inverse transpose Jacobian).
Refutation
By [L1], the cotangent transition matrix is when the tangent transition matrix is .
For the given charts, while , so the tangent and cotangent transition matrices are different.
Therefore the claim is false.
Depends on
Used by
Nothing in the library uses this result yet.
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
- John M. Lee, Introduction to Smooth Manifolds (standard reference, not scraped)
- Will J. Merry, Differential Geometry (standard reference, not scraped)
- Nigel Hitchin, Differentiable Manifolds (standard reference, not scraped)