ExampleConstruction: Literature-sourcedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-29

How statement and proof provenance work

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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The circle from two stereographic charts

Example

Let

$S^{1} : = {(x, y) \in R^{2} : x^{2} + y^{2} = 1} .$

Removing the north pole $N = (0, 1)$ and south pole $S = (0, - 1)$ , define stereographic charts

$σ_{N} (x, y) = \frac{x}{1 - y}, σ_{S} (x, y) = \frac{x}{1 + y} .$

Their inverses are

$σ_{N}^{- 1} (u) = (\frac{2 u}{1 + u^{2}}, \frac{u^{2} - 1}{1 + u^{2}}), σ_{S}^{- 1} (u) = (\frac{2 u}{1 + u^{2}}, \frac{1 - u^{2}}{1 + u^{2}}) .$

These two charts form a smooth atlas on $S^{1}$ , so they exhibit the circle as a smooth $1$ -manifold.

Facts & Assumptions

Given: The circle $S^{1}$ , the poles $N, S$ , and the two stereographic maps $σ_{N}, σ_{S}$ .

[F1]

A smooth atlas is a covering family of pairwise smoothly compatible charts (Smooth atlases).

[F2]

Smooth compatibility requires both transition maps on the overlap to be smooth (Smoothly compatible charts and the smoothness of Euclidean transition maps).

[F3]

A smooth manifold is a topological manifold equipped with a smooth structure (Smooth manifolds and their smooth charts).

Verification

technique · direct

1.1

The domains $S^{1} ∖ {N}$ and $S^{1} ∖ {S}$ are open in $S^{1}$ [given] and cover it. The displayed inverse formulas show that each $σ_{N}$ and $σ_{S}$ is a homeomorphism onto $R$ : composing the inverse with the chart returns the original point, and composing the chart with the inverse gives $u$ . So these are genuine charts.

given

2.1

On the overlap $S^{1} ∖ {N, S}$ the transition maps are

F1F2step 1.1

$σ_{S} \circ σ_{N}^{- 1} (u) = \frac{1}{u}, σ_{N} \circ σ_{S}^{- 1} (u) = \frac{1}{u},$

defined on $R ∖ {0}$ , hence smooth. Therefore the two charts are smoothly compatible by [F2], and [F1] makes them a smooth atlas. [F1, F2, step 1.1]

3.1

This smooth atlas equips $S^{1}$ with a smooth structure, so [F3] makes the [F3, step 2.1] circle a smooth $1$ -manifold.

F3step 2.1∎

Depends on

Used by

The torus as a product smooth manifold Example

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

Rob van der Vorst, Introduction to differentiable manifolds, §1, Example 1.6 (standard reference, not scraped)
Nigel Hitchin, Differentiable Manifolds, §2.3 (standard reference, not scraped)