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

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.

Contracted second Bianchi identity

Statement

This item assumes ${A C}_{ω}$ , namely countable choice. In the propagated dependency chain, that assumption is required through Scalar curvature, Ricci curvature is symmetric and basis independent, and Algebraic symmetries of the Riemann tensor; after those interfaces are fixed, the remaining local or finite argument makes no additional countable-family choice.

For a covariant two-tensor $T$ , write

$(div T) (X) : = \sum_{i} (\nabla_{e_{i}} T) (e_{i}, X)$

in any orthonormal basis at the point. Then

$div Ric = \frac{1}{2} d S,$

and therefore the Einstein tensor is divergence free:

$div (Ric - \frac{1}{2} S g) = 0.$

Facts & Assumptions

[A1]

${A C}_{ω}$ is countable choice and is required here through Scalar curvature, Ricci curvature is symmetric and basis independent, and Algebraic symmetries of the Riemann tensor; after those supplied interfaces are fixed, the remaining local or finite calculation makes no additional countable-family choice.

[F1]

The covariant differential second Bianchi identity is the cyclic sum of $\nabla Rm$ . Differential second Bianchi identity.

[F2]

Scalar curvature is the metric trace of Ricci. Scalar curvature.

[F3]

Ricci curvature is symmetric and has the orthonormal contraction formula. Ricci curvature is symmetric and basis independent.

[F4]

Induced connections commute with permutations and contractions. Induced connections commute with contraction and permutation.

[F5]

Tensor contraction is basis independent. Contraction is independent of the basis formula.

[F6]

The Riemann tensor is skew in both pairs. Algebraic symmetries of the Riemann tensor.

[F7]

The Levi–Civita connection preserves the metric. Levi civita connection.

Proof

Given: ${A C}_{ω}$ , a point $p$ , a vector $X \in T_{p} M$ , and one orthonormal basis $(e_{1}, \dots, e_{n})$ of $T_{p} M$ .

1.1

The displayed definition of $div T$ is a contraction of $\nabla T$ , so [F5] makes it independent of the orthonormal basis. By [F3]–[F4], at $p$ one has $(\nabla_{Y} Ric) (U, V) = \sum_{a} (\nabla_{Y} Rm) (e_{a}, U, V, e_{a})$ . Contracting once more and using [F2] and [F4] gives $d S (X) = \sum_{a, i} (\nabla_{X} Rm) (e_{a}, e_{i}, e_{i}, e_{a})$ .

A1F2F3F4F5

2.1

Insert $(X, e_{a}, e_{i}; e_{i}, e_{a})$ into [F1]'s covariant Bianchi identity and sum over $a, i$ . The first cyclic term is $d S (X)$ by step 1.1. Last-pair skewness in [F6], followed by [F3], turns the second term into $- \sum_{a} (\nabla_{e_{a}} Ric) (e_{a}, X)$ . First-pair skewness followed by [F3] turns the third into the same expression with index $i$ . Hence $d S (X) - 2 (div Ric) (X) = 0$ .

F1F3F6step 1.1algebra

3.1

Let $G = Ric - (S / 2) g$ . Metric compatibility [F7] and an orthonormal expansion give $div (S g) (X) = \sum_{i} e_{i} (S) g (e_{i}, X) = d S (X)$ . Therefore step 2.1 yields $div G = div Ric - (1 / 2) d S = 0$ .

F7step 2.1algebra∎

Depends on

Used by

Schur's lemma for pointwise constant sectional curvature Theorem

Dependency tree · two levels

27 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)