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.

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The contraction of a mixed tensor

Definition

Let $T$ be a type $(r, s)$ tensor on $V$ with $r, s \geq 1$ . Contracting the first contravariant slot against the first covariant slot gives the type $(r - 1, s - 1)$ tensor $ctr (T)$ defined intrinsically by

$ctr (T) (α_{2}, \dots, α_{r}, v_{2}, \dots, v_{s}) : = \sum_{i} T (e^{i}, α_{2}, \dots, α_{r}, e_{i}, v_{2}, \dots, v_{s}),$

where $(e_{i})$ is any basis of $V$ and $(e^{i})$ its dual basis. The next lemma shows that this formula is basis-independent.

Depends on

A type $(r,s)$ tensor on a finite-dimensional vector space

Used by

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

Will J. Merry, Differential Geometry (standard reference, not scraped)