Pseudo-Riemannian Foliations and Their Graphs

Abstract

We prove that a foliation (M,F) of codimension q on a n-dimensional pseudo-Riemannian manifold with induced metrics on leaves is pseudo-Riemannian if and only if any geodesic that is orthogonal at one point to a leaf is orthogonal to every leaf it intersects. We show that on the graph G = G(F) of a pseudo-Riemannian foliation there exists a unique pseudo-Riemannian metric such that canonical projections are pseudo-Riemannian submersions and the fibers of different projections are orthogonal at common points. Relatively this metric the induced foliation (G, F) on the graph is pseudo-Riemannian and the structure of the leaves of (G, F) is described. Special attention is given to the structure of graphs of transversally (geodesically) complete pseudo-Riemannian foliations which are totally geodesic pseudo-Riemannian ones.