Closed vector fields

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. 1* Notation* Let M be a connected C semi-Riemannian manifold with metric tensor <^ , )> and Riemannian connexion D [see Helgason 4 or Hicks 7 for definitions]. We summarize the properties of D and some associated concepts we shall use. The operator D assigns to each pair of C vector fields X and Y on an open set U of M, a C vector field D x Y called the covariant derivative of Y in the direction X. If X, Y, and Z are C fields on U and / a C function (real valued) on U then we have the following relations between vector fields on U: