TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14

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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.

Algebraic symmetries of the Riemann tensor

Statement

This item assumes ${A C}_{ω}$ , namely countable choice. In the propagated dependency chain, that assumption is required through First Bianchi identity; after those interfaces are fixed, the remaining local or finite argument makes no additional countable-family choice.

The Riemann curvature four-tensor satisfies, for all vector fields $X, Y, Z, W$ ,

$Rm (X, Y, Z, W) = - Rm (Y, X, Z, W),$

$Rm (X, Y, Z, W) = - Rm (X, Y, W, Z),$

$Rm (X, Y, Z, W) = Rm (Z, W, X, Y),$

and

$Rm (X, Y, Z, W) + Rm (Y, Z, X, W) + Rm (Z, X, Y, W) = 0.$

These are respectively first-pair skewness, last-pair skewness, pair interchange, and the cyclic first-Bianchi symmetry.

Facts & Assumptions

[A1]

${A C}_{ω}$ is countable choice and is required here through First Bianchi identity; after those supplied interfaces are fixed, the remaining local or finite calculation makes no additional countable-family choice.

[F1]

$Rm (X, Y, Z, W) = g (R (X, Y) Z, W)$ . Riemann curvature four-tensor.

[F2]

Curvature is skew in its first two arguments. Curvature is skew in its first two arguments.

[F3]

Curvature obeys the cyclic first Bianchi identity. First Bianchi identity.

[F4]

The Levi–Civita connection is metric compatible. Levi civita connection.

Proof

Given: ${A C}_{ω}$ , smooth vector fields $X, Y, Z, W$ and the Levi–Civita connection.

1.1

Combining [F1] with [F2] gives first-pair skewness, and pairing [F3] with $W$ gives the displayed cyclic identity for $Rm$ .

A1F1F2F3

1.2

Metric compatibility [F4] expands the scalar identity $X Y g (Z, W) - Y X g (Z, W) - [X, Y] g (Z, W) = 0$ . The mixed terms $g (\nabla_{Y} Z, \nabla_{X} W)$ and $g (\nabla_{X} Z, \nabla_{Y} W)$ cancel in pairs, leaving $g (R (X, Y) Z, W) + g (Z, R (X, Y) W) = 0$ . Symmetry of $g$ and [F1] give last-pair skewness.

F1F4algebra

2.1

Write the cyclic identity from step 1.1 for the four ordered triples $(X, Y, Z; W)$ , $(Y, Z, W; X)$ , $(Z, W, X; Y)$ , and $(W, X, Y; Z)$ and add them. Last-pair skewness from step 1.2 cancels the eight terms whose first pair is respectively $(X, Y)$ , $(Y, Z)$ , $(Z, W)$ , or $(W, X)$ . The four remaining terms, simplified with both pair skews, give $2 Rm (Y, W, X, Z) - 2 Rm (X, Z, Y, W) = 0$ . Renaming $(X, Z, Y, W)$ as an arbitrary quadruple yields pair interchange.

step 1.1step 1.2algebra∎

Depends on

Used by

Sectional curvature Definition
Ricci curvature and scalar curvature determine the full Riemann tensor in every dimension False statement
Ricci curvature is symmetric and basis independent Lemma
Sectional curvature is independent of the basis of the plane Lemma
Ricci decomposition of the Riemann tensor in dimension at least three Proposition
Scalar curvature is twice the sum of sectional curvatures of orthonormal coordinate planes Proposition
Contracted second Bianchi identity Theorem

Dependency tree · two levels

19 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

Ved Datar, Lectures on Riemannian Geometry (standard reference, not scraped)
John M. Lee, Riemannian Manifolds: An Introduction to Curvature (standard reference, not scraped)