ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01

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A coordinate-bump embedding of the circle in Euclidean space

Example

Cover $S^{1}$ by the two standard stereographic-coordinate charts

U_{+} : = S^{1} ∖ {(- 1, 0)}, U_{-} : = S^{1} ∖ {(1, 0)},

with angular coordinates $θ_{+}$ and $θ_{-}$ . Choose smooth bumps $ϕ_{\pm}$ supported in $U_{\pm}$ and equal to $1$ on smaller arcs still covering $S^{1}$ . Then

F (p) : = (ϕ_{+} (p), ϕ_{+} (p) θ_{+} (p), ϕ_{-} (p), ϕ_{-} (p) θ_{-} (p))

is an explicit coordinate-bump embedding of $S^{1}$ into $R^{4}$ .

Facts & Assumptions

Given: The two-chart cover of $S^{1}$ and the chosen bump functions $ϕ_{\pm}$ .

Verification

technique · direct

1.1

On any point of the smaller arc where $ϕ_{+} = 1$ , the first two coordinates of $F$ recover the chart coordinate $θ_{+}$ . The same is true for $ϕ_{-} = 1$ on the other smaller arc.

givenalgebra

2.1

Because those smaller arcs cover $S^{1}$ , equality $F (p) = F (q)$ forces equality in one active chart coordinate, hence $p = q$ . The active chart coordinates also show that $d F$ is injective at each point. This is exactly the compact coordinate-bump mechanism of A finite coordinate-bump map embeds a compact manifold in some Euclidean space.

step 1.1∎

Depends on

A finite coordinate-bump map embeds a compact manifold in some Euclidean space

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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

Marco Gualtieri, Topology I: Smooth Manifolds, Part 11 (standard reference, not scraped)