I see this in general relativity, and it is "locally synchronizable" implies "synchronizable" locally, where we say a reference frame $Q$ (future-point vector field such that $g(Q,Q)=-1$) is "locally synchronizable" if the associated 1-form $\omega$ satisfies $\omega\wedge \text{d}\omega=0$ and $Q$ is "synchronizable" if there exist smooth functions such that $\omega=h\text{d}f$.
I can not find a proof for this.
Is it true for general (semi-)Riemannian manifold?
Thank you for any suggestions.