Poincar\'e lemmaPoincareLemma2004-01-11 06:25:362007-02-28 09:33:13Theorem% this is the default PlanetMath preamble. as your knowledge
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The Poincar\'e lemma states that every closed differential form
is locally \PMlinkname{exact}{ExactDifferentialForm}.
\begin{theorem*} (Poincar\'e Lemma)
\cite{conlon} 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 $k>0$.
\begin{itemize}
\item
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 $\iota:U\hookrightarrow X$.
\item If $X$ is contractible, this $\eta$ exists globally; there exists a
$(k-1)$-form $\eta\in \Omega^{k-1}(X)$ such that
$$ d\eta = \omega.$$
\end{itemize}
\end{theorem*}
\subsubsection*{Notes}
Despite the name, the Poincar\'e 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 Poincar\'e 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 $X=\sR^2\setminus\{0\}$ with polar coordinates $(r,\phi)$,
the $1$-form $\omega=d\phi$ is not globally exact.
\begin{thebibliography}{9}
\bibitem {conlon} L. Conlon, \emph{Differentiable Manifolds: A first course},
Birkh\"auser, 1993.
\end{thebibliography}