Closed differential one-form ($\eta \in {\Omega }^1(M),d\eta =0$)

A differential one-form is closed when its exterior derivative vanishes. In coordinates, $\eta ={\eta }_{i}d{x}^i$ is closed exactly when ${\partial }_i{\eta }_j={\partial }_j{\eta }_i$. The Poincare lemma makes it locally an exact differential form, but global exactness can fail through nonzero integrals around closed curves.