ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-13

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.

Straight lines as Euclidean geodesics

Example

On Euclidean $R^{n}$ with its Levi–Civita connection, every affinely parametrized geodesic on an interval $I$ has the form $γ (t) = p + t v (t \in I)$ for fixed $p, v \in R^{n}$ , and every such curve is a geodesic. Here $v = 0$ gives a constant geodesic; a nonconstant curve traces a straight line. If $n = 0$ , the only curves are constant.

Facts & Assumptions

Given: The Euclidean metric $g_{i j} = δ_{i j}$ in Cartesian coordinates and an interval $I$ of affine parameter values.

[F1]

Christoffel formula for the levi civita connection gives $Γ^{k}_{i j} = \frac{1}{2} g^{k ℓ} (\partial_{i} g_{j ℓ} + \partial_{j} g_{i ℓ} - \partial_{ℓ} g_{i j})$ .

[F2]

Coordinate geodesic equation says that $γ$ is a geodesic exactly when ${\ddot{x}}^{k} + Γ^{k}_{i j} {\dot{x}}^{i} {\dot{x}}^{j} = 0$ in every coordinate.

[F3]

A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$ ; consequently two such functions with the same derivative differ by a constant says that a continuous function on an interval whose interior derivative is zero is constant, including when the interval has endpoints.

Verification

1.1

Every $g_{i j} = δ_{i j}$ is constant, so all its partial derivatives vanish. Formula [F1] therefore gives $Γ^{k}_{i j} = 0$ for every index.

F1given

2.1

By [F2] and step 1.1, the geodesic equation is ${\ddot{x}}^{k} = 0$ for each $k$ . Applying [F3] first to ${\dot{x}}^{k}$ gives a constant $v^{k}$ ; applying it to $x^{k} (t) - t v^{k}$ gives a constant $p^{k}$ . Thus $γ (t) = p + t v$ throughout the interval, not merely near one parameter value. For an included endpoint the equality extends by continuity.

F2F3step 1.1

3.1

Conversely, $x^{k} (t) = p^{k} + t v^{k}$ has ${\ddot{x}}^{k} = 0$ , so [F2] and step 1.1 make it a geodesic. If $v = 0$ , it is constant; if $v \neq 0$ , its image lies on the straight line $p + R v$ . In dimension zero there are no coordinate equations and the unique curve is constant. The argument makes no choice beyond the given curve's own coordinates.

F2step 1.1step 2.1∎

Depends on

Used by

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Dependency tree · two levels

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Sources

Ved Datar, Lectures on Riemannian Geometry, Example 15.1.3, p. 114 (standard reference, not scraped)