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

The tangent bundle as a disjoint union

Definition

Depends on

Dependency tree · 2 levels

Sources