We study closed vector fields on a semi-Riemannian manifold. In particular, we study
the differential geometry of the submanifolds determined by a nonvanishing closed
field. Expressions are computed for the Weingarten map, the mean curvature, the
Riemannian curvature, and the Laplacian of the square of the length of the
field. Thus we obtain a necessary and sufficient condition that the constant
hypersurface of a nontrivial harmonic function be a minimal surface. We obtain
conditions that imply the classical Codazzi-Mainardi equations hold. We obtain
conditions that imply the existence of a representation of the manifold as a cross
product in which one factor is a real line. Finally, various special cases are
examined.
