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

1.1F1given

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.

2.1F2F3step 1.1

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.

3.1F2step 1.1step 2.1∎

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.

Depends on

Used by

Corner terms are required even in the plane Counterexample
Distance hessian in euclidean space Example
Flat torus model geometry Proposition

Dependency tree · two levels

17 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, Example 15.1.3, p. 114 (standard reference, not scraped)