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, ${\mathrm{\Omega }}^k\mathit{}\mathrm{(}X\mathrm{)}$ is the set of smooth differential $k$-forms on $X$, and suppose $\omega$ is a closed form in ${\mathrm{\Omega }}^k\mathit{}\mathrm{(}X\mathrm{)}$ 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 an extremely 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 not globally exact. For instance, in $X={ℝ}^2\setminus \{0\}$ with polar coordinates $(r,\varphi )$, the $1$-form $\omega =d\varphi$ is not globally exact.

Title	Poincaré lemma
Canonical name	PoincareLemma
Date of creation	2013-03-22 14:06:28
Last modified on	2013-03-22 14:06:28
Owner	matte (1858)
Last modified by	matte (1858)
Numerical id	12
Author	matte (1858)
Entry type	Theorem
Classification	msc 53-00
Related topic	ExactDifferentialForm
Related topic	ClosedDifferentialFormsOnASimpleConnectedDomain
Related topic	LaminarField