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.

Cotangent space and cotangent bundle as a disjoint union

Definition

For a smooth manifold $M$ and a point $p \in M$ , the cotangent space at $p$ is the algebraic dual $T_{p}^{*} M : = (T_{p} M)^{*}$ . The cotangent bundle is the disjoint union $T^{*} M : = ⨆_{p \in M} T_{p}^{*} M .$

Cotangent space and cotangent bundle as a disjoint union

Definition

Depends on

Dependency tree · 2 levels

Sources