= Normalized Killing one-form
{title2=$\eta=V^\flat/g(V,V)$}
For a non-null <Killing vector field> $V$ of a <Levi-Civita connection>, <hypersurface orthogonality> implies that $\eta=V^\flat/F$, with $F=g(V,V)$, is a <closed differential form>. The <Killing equation> and $V^\flat\wedge dV^\flat=0$ give $2F\nabla_\mu V_\nu=V_\nu\partial_\mu F-V_\mu\partial_\nu F$, whence $d\eta=0$. The <Poincare lemma> gives $\eta=d\phi$ locally. For a timelike $V$, $\phi$ supplies local static time coordinates. This does not assert global exactness.
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