DefinitionDefinition: AI-adaptedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-30

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.

The tangent bundle as a disjoint union

Definition

The tangent bundle of a smooth manifold $M$ is the disjoint union $T M : = ⨆_{p \in M} T_{p} M .$ An element of $T M$ is a tangent vector together with its base point, and the projection $π : T M \to M$ sends a vector in $T_{p} M$ to $p$ . At this stage, no topology or smooth structure is yet imposed on $T M$ .

Depends on

Derivations at a point and the tangent space

Used by

Cotangent space and cotangent bundle as a disjoint union Definition
The global differential or tangent map Definition
The induced tangent bundle chart Definition
Assuming countable choice, the tangent bundle has a canonical smooth 2n-manifold structure Theorem

Dependency tree · two levels

3 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)