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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

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 $i, j, k$ .

1.1F2algebra

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

2.1F1F3step 1.1algebra∎

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.

Depends on

Used by

A cut point that is not conjugate because two minimizers arrive Counterexample
Ricci lower bound does not control every sectional curvature in dimension at least three Counterexample
A deformed sphere has the same total curvature Example
A flat torus showing simple connectedness is needed for global exp injectivity Example
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
Jacobi fields in euclidean space Example
A geodesic stops minimizing exactly at its first conjugate point False statement
Cartan hadamard says exp p is injective without simple connectedness False statement
Christoffel symbols vanishing at one point implies curvature vanishes there False statement
Every vector field along a geodesic is a Jacobi field False statement
Ricci curvature and scalar curvature determine the full Riemann tensor in every dimension False statement
Flat torus model geometry Proposition
Upper half-space model geometry Proposition

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)