Graphs of totally geodesic foliations on pseudo-Riemannian manifolds

Abstract

We study totally geodesic foliations $(M, F)$ of arbitrary codimension $q$ on $n$-dimensional pseudo-Riemannian manifolds, for which the induced metrics on leaves is non-degenerate. We assume that the $q$-dimensional orthogonal distribution $\mathfrak{M}$ to $(M, F)$ is an Ehresmann connection for this foliation. Since the usual graph $G(F)$ is not Hausdorff manifold in general, we study the graph $G_{\mathfrak{M}}(F)$ of the foliation with an Ehresmann connection $\mathfrak M$ introduced early by the author. This graph is always a Hausdorff manifold. We prove that on the graph $G_{\mathfrak{M}}(F)$, a pseudo-Riemannian metric is defined, with respect to which the induced foliation and the simple foliations formed by the leaves of the canonical projections are totally geodesic. We show that the leaves of the induced foliation on the graph are non-degenerately reducible pseudo-Riemannian manifolds and their structure is described. The application to parallel foliations on nondegenerately reducible pseudo-Riemannian manifolds is considered. We also show that each foliation defined by the suspension of a homomorphism of the fundamental group of a pseudo-Riemannian manifold belongs to the considered class of foliations.