Closed Ernst contraction and the real potential

Each term in the invariant formula [F9] is real-linear in the form, so $d(aU+bV)=adU+bdV$ for real constants $a,b$. Consequently the componentwise extension in [F9] is complex-linear and commutes with taking real and imaginary parts. Applying the real Cartan identity separately to those parts gives [F7] for complex forms. By [F3]–[F5], $d\mathcal{F}=dF+id{\star }_{g}F=0$. Applying [F7] and then [F6] gives $d\eta =2({\mathcal{L}}_T\mathcal{F}-{\iota }_{T}d\mathcal{F})=0$. By [F10], metric compatibility and the antisymmetry of $F$ give $(df{)}_b=-2T^a{\mathrm{\nabla }}_b{T}_a=2T^a{F}_{ab}=\mathrm{Re}{\eta }_b$.

Restrict to a coordinate ball; its image is star-shaped. By step 1.1, both real and imaginary parts of $\eta$ are closed smooth real one-forms. Evaluating the invariant formula [F9] on commuting coordinate vector fields identifies closedness of either real one-form with ${\mathrm{\partial }}_j{A}_k={\mathrm{\partial }}_k{A}_j$ for its real coefficient field $A$. Thus their coefficient fields meet the closed $C^1$ hypotheses of [F8], which supplies real potentials whose complex combination is a local $\sigma$. Their first partial derivatives are smooth components of $\eta$, so these potentials are smooth. Wherever a global primitive is assumed, step 1.1 and [F9]–[F10] give $d(\mathrm{Re}\sigma -f)=0$. This difference is constant on each coordinate ball by integrating its zero derivative along segments, and hence locally constant. On each connected component it is constant, since every nonempty fiber of a locally constant function is open and closed. No assertion that all periods vanish is used.