Poincaré lemma

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

Theorem.

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

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

  $$d\eta =\omega .$$

Notes

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

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

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 notglobally exact. For instance, in $X={ℝ}^2\setminus \{0\}$ with polar coordinates $(r,\varphi )$,the $1$-form $\omega =d\varphi$ is not globally exact.