Poincare lemma

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

Suppose $X$ is a smooth manifold, ${\mathrm{\Omega }}^k(X)$ is the set of smooth differential $k$-forms on $X$, and suppose $\omega$ is a closed form in ${\mathrm{\Omega }}^k(X)$ for some $k>0$. Then for every $x\in X$ there is a neighbourhood $U\subset X$, and a $(k-1)$-form $\eta \in {\mathrm{\Omega }}^{k-1}(U)$ , such that $d\eta =i^{\ast }\omega$, where $i$ is the inclusion $i:U\hookrightarrow X$.

If $X$ is a contractible space, this $\eta$ exists globally; there exists a $(k-1)$-form $\eta \in {\mathrm{\Omega }}^{k-1}(X)$ such that $d\eta =\omega$.

For $1$-forms, you only need $X$ to be simply connected.

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Author of the notes: Antonio J. Pan-Collantes

antonio.pan@uca.es