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.
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- 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.
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Every smooth vector field on a compact manifold is complete
Statement
Assume (The Axiom of Countable Choice ()). Every smooth vector field on a compact manifold is complete.
Facts & Assumptions
Given: , a compact smooth manifold , and a smooth vector field on .
Under , a compactly supported smooth vector field is complete (Compactly supported smooth vector fields are complete). This is the exact use of the choice assumption.
The support of a section is a closed subset of the base manifold (Smooth sections, local sections, and support).
Closed subsets of compact spaces are compact (A closed subspace of a compact space is compact, and a finite union of compact subspaces is compact).
Proof
By [L2], the support of is a closed subset of . Since is compact, [L3] shows that is compact.
Thus is compactly supported, so [L1] implies that is complete.
Depends on
- The Axiom of Countable Choice ($\mathrm{AC}_\omega$)
- Compactly supported smooth vector fields are complete
- Smooth sections, local sections, and support
- Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right
- A closed subspace of a compact space is compact, and a finite union of compact subspaces is compact
Used by
- Conformal removability of compact sets Definition
- Broken trajectories in an index-two torus moduli space Example
- The Morse complex of the circle Example
- The Morse complex of the two-sphere Example
- A compact holonomy-free codimension-one foliation is fibered over its leaf space Lemma
- A negative-gradient trajectory on a compact Morse manifold has single critical alpha and omega limits Lemma
- Continuation solutions have critical limits and exponential decay Lemma
- Stabilizer of the ℝⁿ-action on a compact connected regular fibre is a full lattice Lemma
- A transverse volume-preserving flow implies tautness in the compact cooriented three-dimensional setting Proposition
- Every Morse function admits a complete downward gradient-like field on a closed manifold Proposition
- Compactness up to breaking needs closedness or a proper compactness package Remark
- Liouville–Arnold action–angle theorem Theorem
Dependency tree · two levels
22 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
- Will J. Merry, Differential Geometry (standard reference, not scraped)
- Nigel Hitchin, Differentiable Manifolds (standard reference, not scraped)