DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 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 pullback of a covariant tensor by a linear map

Definition

Let $V$ and $W$ be finite-dimensional real vector spaces, let $k \geq 0$ be an integer, let $A : V \to W$ be linear, and let $T : W^{k} \to R$ be a covariant tensor of degree $k$ . Its pullback $A^{*} T$ is the covariant tensor on $V$ defined by

(A^{*} T) (v_{1}, \dots, v_{k}) : = T (A v_{1}, \dots, A v_{k}) .

Depends on

Used by

Linear pullback respects tensor products and permutations Proposition

Dependency tree · two levels

5 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, 2nd ed. (standard reference, not scraped)