ExampleConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedprecheck passaudited 2026-09-07

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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.
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The product orientation on a torus

Example

The product orientation on $S^{1} \times S^{1}$ is the ray of the ordered pair of positive tangent directions.

Facts & Assumptions

Given: Each factor $S^{1}$ with its chosen orientation, positively oriented angular coordinates $θ_{1}$ and $θ_{2}$ , and the factor order $S^{1} \times S^{1}$ .

[L1]

The product orientation is the tensor product of the two determinant rays under the ordered splitting of the tangent space (Product orientations).

[L2]

An oriented chart is one whose ordered coordinate frame lies in the selected determinant ray (Oriented smooth manifolds and oriented charts).

Verification

technique · direct

1.1

At $(p_{1}, p_{2})$ , the ordered tangent-space splitting is $T_{p_{1}} S^{1} \oplus T_{p_{2}} S^{1}$ . By [L1], the product ray is generated by placing a positive vector of the first factor before a positive vector of the second.

givenL1algebra

2.1

The angular product chart has ordered frame $(\partial_{θ_{1}}, \partial_{θ_{2}})$ , whose entries are positive in their respective circle factors. Step 1.1 and [L2] therefore identify its ray with the product orientation, which is the standard torus orientation for these chosen angular directions and factor order.

givenL2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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

Ioan Mărcuț, Manifolds (2017 lecture notes), §§14.5, 15.1 (standard reference, not scraped)
Will Merry, Differential Geometry (2021), Lecture 24 (standard reference, not scraped)