PropositionStatement: 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.

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Coordinate formula for the curvature tensor

Statement

In a coordinate chart, use the convention

$\nabla_{\partial_{i}} \partial_{k} = Γ^{ℓ}_{i k} \partial_{ℓ}, R (\partial_{i}, \partial_{j}) \partial_{k} = R^{ℓ}_{k i j} \partial_{ℓ} .$

Then

$R^{ℓ}_{k i j} = \partial_{i} Γ^{ℓ}_{j k} - \partial_{j} Γ^{ℓ}_{i k} + Γ^{m}_{j k} Γ^{ℓ}_{i m} - Γ^{m}_{i k} Γ^{ℓ}_{j m} .$

Facts & Assumptions

[F1]

Curvature is $R (X, Y) Z = \nabla_{X} \nabla_{Y} Z - \nabla_{Y} \nabla_{X} Z - \nabla_{[X, Y]} Z$ . Curvature of an affine connection.

[F2]

The Christoffel symbols satisfy $\nabla_{\partial_{i}} \partial_{j} = Γ^{k}_{i j} \partial_{k}$ , with the first lower index the differentiating direction. Christoffel symbols of an affine connection.

[F3]

Coordinate vector fields commute. Coordinate vector fields commute.

Proof

Given: A coordinate chart $(x^{1}, \dots, x^{n})$ and indices $i, j, k$ .

1.1

Applying the connection Leibniz rule to [F2] twice gives $\nabla_{\partial_{i}} \nabla_{\partial_{j}} \partial_{k} = (\partial_{i} Γ^{ℓ}_{j k} + Γ^{m}_{j k} Γ^{ℓ}_{i m}) \partial_{ℓ}$ and, after interchanging $i, j$ , $\nabla_{\partial_{j}} \nabla_{\partial_{i}} \partial_{k} = (\partial_{j} Γ^{ℓ}_{i k} + Γ^{m}_{i k} Γ^{ℓ}_{j m}) \partial_{ℓ}$ .

F2algebra

2.1

By [F3], the bracket term in [F1] is zero. Subtracting the two expansions from step 1.1 therefore makes the coefficient of $\partial_{ℓ}$ exactly $\partial_{i} Γ^{ℓ}_{j k} - \partial_{j} Γ^{ℓ}_{i k} + Γ^{m}_{j k} Γ^{ℓ}_{i m} - Γ^{m}_{i k} Γ^{ℓ}_{j m}$ , as claimed.

F1F3step 1.1algebra∎

Depends on

Used by

Curvature of a Riemannian product Example
Euclidean space has zero curvature Example
Gaussian curvature of a surface of revolution Example
Hyperbolic space has negative constant sectional curvature Example
Christoffel symbols vanishing at one point implies curvature vanishes there False statement
Ricci curvature and scalar curvature determine the full Riemann tensor in every dimension False statement

Dependency tree · two levels

7 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)