Poincaré lemma

The Poincaré lemma states that every closed differential form is locally exact (http://planetmath.org/ExactDifferentialForm).

Theorem.

(Poincaré Lemma) [1] Suppose $X$ is a smooth manifold, $\Omega^{k}(X)$ is the set of smooth differential $k$-forms on $X$, and suppose $\omega$ is a closed form in $\Omega^{k}(X)$ for some $k>0$.

• Then for every $x\in X$ there is a neighbourhood $U\subset X$, and a $(k-1)$-form $\eta\in\Omega^{k-1}(U)$, such that

 $d\eta=\iota^{\ast}\omega,$

where $\iota$ is the inclusion $\iota:U\hookrightarrow X$.

• If $X$ is contractible, this $\eta$ exists globally; there exists a $(k-1)$-form $\eta\in\Omega^{k-1}(X)$ such that

 $d\eta=\omega.$

Notes

Despite the name, the Poincaré lemma is an extremely important result. For instance, in algebraic topology, the definition of the $k$th de Rham cohomology group

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

can be seen as a measure of the degree in which the Poincaré lemma fails. If $H^{k}(X)=0$, then every $k$ form is exact, but if $H^{k}(X)$ is non-zero, then $X$ has a non-trivial topology (or “holes”) such that $k$-forms are not globally exact. For instance, in $X=\mathbb{R}^{2}\setminus\{0\}$ with polar coordinates $(r,\phi)$, the $1$-form $\omega=d\phi$ is not globally exact.

References

• 1 L. Conlon, Differentiable Manifolds: A first course, Birkhäuser, 1993.
Title Poincaré lemma PoincareLemma 2013-03-22 14:06:28 2013-03-22 14:06:28 matte (1858) matte (1858) 12 matte (1858) Theorem msc 53-00 ExactDifferentialForm ClosedDifferentialFormsOnASimpleConnectedDomain LaminarField