False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30

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Weighted sums do not glue arbitrary manifold-valued maps

Statement

False claim: a partition of unity can glue arbitrary manifold-valued maps by weighted sums of their values.

Facts & Assumptions

Given: The target manifold $S^{1} \subseteq R^{2}$ , the two-member cover $U = V = R$ , the subordinate smooth partition $ϕ_{U} = ϕ_{V} = 1 / 2$ , and the constant maps $f_{U} \equiv (1, 0)$ and $f_{V} \equiv (- 1, 0)$ .

[L1]

Partition-of-unity gluing works for real-valued functions because addition and scalar multiplication are available in the target (Smooth locally defined functions can be glued by a partition of unity).

[A1]

On the overlap $U \cap V = R$ , the partition weights are $1 / 2$ and $1 / 2$ .

Refutation

technique · direct

1.1

On the overlap, the weighted sum would be $(1 / 2) f_{U} + (1 / 2) f_{V} = (0, 0)$ .

A1given

2.1

The point $(0, 0)$ does not lie on $S^{1}$ , so the weighted sum leaves the manifold target.

step 1.1

3.1

Thus the affine argument from [L1] does not extend to arbitrary manifold-valued maps.

L1step 2.1∎

Depends on

Smooth locally defined functions can be glued by a partition of unity

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

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)