DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-01

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 type $(r, s)$ tensor bundle

Definition

Let $M$ be a smooth manifold, and let $r, s \geq 0$ . The type $(r, s)$ tensor bundle of $M$ is the fibrewise assignment

$T_{s}^{r} M : = ∐_{p \in M} Mult ((T_{p}^{*} M)^{r} \times (T_{p} M)^{s}, R),$

that is, the fibre over $p$ consists of the type $(r, s)$ tensors on $T_{p} M$ in the multilinear-map sense. The next theorem equips this disjoint union with its smooth vector-bundle structure.

The type $(r, s)$ tensor bundle

Definition

Depends on

Dependency tree · 2 levels

Sources

The type (r,s)(r,s) tensor bundle

Definition

Depends on

Dependency tree · 2 levels

Sources

The type $(r, s)$ tensor bundle