PlanetMath: Poincaré lemma

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Poincaré lemma

(Theorem)

The Poincaré lemma states that every closed differential form is locally exact.

Theorem (Poincaré Lemma) [1] Suppose is a smooth manifold, $\Omega^k(X)$ is the set of smooth differential -forms on , and suppose $\omega$ is a closed form in $\Omega^k(X)$ for some .

Then for every $x\in X$ there is a neighbourhood $U\subset X$ , and a -form $\eta\in \Omega^{k-1}(U)$ , such that
$\displaystyle d\eta = \iota^\ast \omega,$
where $\iota$ is the inclusion $\iota:U\hookrightarrow X$ .
If is contractible, this $\eta$ exists globally; there exists a -form $\eta\in \Omega^{k-1}(X)$ such that
$\displaystyle d\eta = \omega.$

Notes

Despite the name, the Poincaré lemma is an extremely important result. For instance, in algebraic topology, the definition of the

th de Rham cohomology group

$\displaystyle H^k(X) = \frac{ \operatorname{Ker}\{ d\colon \Omega^k(X)\to \Omega^{k+1}(X)\}}{ \operatorname{Im}\{ d\colon \Omega^{k-1}(X)\to \Omega^{k}(X)\}}$