ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-08-30

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.

The tangent bundle of the circle is a cylinder

Example

The tangent bundle of $S^{1}$ is diffeomorphic to the cylinder $S^{1} \times R$ . If $p = (x, y)$ and $q = (- y, x)$ , the diffeomorphism sends $(p, a)$ to the tangent derivation represented by $γ_{p, a} (t) : = \cos (a t) p + \sin (a t) q .$ Under the canonical ambient-velocity identification, this tangent vector is $a (- y, x)$ .

Facts & Assumptions

Given: A point $p = (x, y) \in S^{1}$ and a scalar $a \in R$ .

[L1]

The tangent bundle is a smooth manifold whose fibers are the tangent spaces (Assuming countable choice, the tangent bundle has a canonical smooth 2n-manifold structure).

[L2]

Curve contact classes are canonically identified with derivation tangent vectors (Curve contact classes are canonically isomorphic to derivation tangent vectors).

[L3]

A smooth coordinate chart induces tangent-bundle coordinates by recording the coefficients in its coordinate tangent basis (The induced tangent bundle chart).

Verification

technique · direct

1.1

The curve $γ_{p, a}$ lies in $S^{1}$ , starts at $p$ , and has ambient derivative $a (- y, x)$ at $0$ . By [L2] it therefore determines a tangent derivation at $p$ .

givenL2algebra

2.1

Let $p_{0} \in S^{1}$ and choose a local angle chart $θ \mapsto p (θ) = (\cos θ, \sin θ)$ around $p_{0}$ . Its coordinate tangent vector has ambient velocity $(- \sin θ, \cos θ)$ , so [L2] makes every tangent derivation over this arc uniquely $a \partial_{θ}$ . In the induced chart of [L3], the displayed map is therefore $(θ, a) ⟼ (θ, a) .$ Thus it and its inverse are smooth on every such bundle-chart domain.

L1L2L3step 1.1

3.1

Hence $T S^{1}$ is a cylinder.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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

John M. Lee, Introduction to Smooth Manifolds (standard reference, not scraped)
Will J. Merry, Differential Geometry (standard reference, not scraped)
Nigel Hitchin, Differentiable Manifolds (standard reference, not scraped)