TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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Poincare lemma for differential forms on star shaped domains

Statement

Every closed smooth $k$ -form on a star-shaped open domain is exact for $k \geq 1$ . For centre $0$ , one primitive is $η_{x} (v_{1}, \dots, v_{k - 1}) = \int_{0}^{1} t^{k - 1} ω_{t x} (x, v_{1}, \dots, v_{k - 1}) d t$ .

Facts & Assumptions

Given: A domain $U$ star-shaped about $c$ , and a closed $k$ -form $ω$ with $k \geq 1$ .

[F1]

Radial contraction of a star shaped domain: For an open $U \subseteq R^{n}$ star-shaped about a specified $c \in U$ , the radial contraction is $F : U \times [0, 1] \to U$ , $F (x, t) = c + t (x - c)$ . def-star-shaped-open-subset-of-rn says exactly that each displayed value lies in $U$ . The coordinate expression is polynomial, so its restriction is smooth up to both endpoints; $F (x, 0) = c$ and $F (x, 1) = x$ . The centre is part of the data, so $U$ is nonempty. For $n = 0$ the unique nonempty domain is a point and the formula is constant.

[F2]

De rham homotopy formula for a smooth homotopy: If $F : M \times [0, 1] \to N$ is smooth up to the endpoints and $F_{t} (x) = F (x, t)$ , then $F_{1}^{*} - F_{0}^{*} = d (K F^{*}) + (K F^{*}) d$ .

Proof

technique · direct

1.1

Take the radial homotopy from the constant map to the identity. Its time-zero pullback on positive-degree forms vanishes because the differential of the constant map is zero. The homotopy formula and $d ω = 0$ give $ω = d (K F^{*} ω)$ , so $η = K F^{*} ω$ is a smooth primitive.

F1F2given

2.1

For $c = 0$ , $d F_{(x, t)} \partial_{t} = x$ and $d F_{(x, t)} (v, 0) = t v$ . The contraction coefficient therefore equals $t^{k - 1} ω_{t x} (x, v_{1}, \dots, v_{k - 1})$ . Integrating gives the displayed formula. When $k = 1$ the factor is $1$ , including $t = 0$ ; for $k > 1$ the integrand is smooth and vanishes there. Degrees exceeding the dimension have zero form and zero primitive.

F2step 1.1∎

Source locator

Lee, Theorem 17.14, p.447; the explicit primitive follows by evaluating the interval operator.

Depends on

Used by

Closed differential forms are locally exact Corollary
De rham cohomology of punctured euclidean space Corollary
De rham cohomology of euclidean space Example
The poincare lemma says every closed form is globally exact False statement
De rham cohomology of spheres Theorem

Dependency tree · two levels

8 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, second edition (standard reference, not scraped)
Nigel Hitchin, Differentiable Manifolds (2014) (standard reference, not scraped)