Pseudo-Riemannian foliations and their graphs

TL;DR

This paper characterizes pseudo-Riemannian foliations on pseudo-Riemannian manifolds via orthogonality of geodesics and constructs a unique compatible metric on their graphs, analyzing the structure of leaves and special cases.

Contribution

It provides a characterization of pseudo-Riemannian foliations through geodesic orthogonality and establishes a unique metric on their graphs with detailed structural analysis.

Findings

Geodesic orthogonality characterizes pseudo-Riemannian foliations.

Existence of a unique pseudo-Riemannian metric on the graph of a foliation.

Structural description of leaves in the graph and special cases.

Abstract

We prove that a foliation $(M, F)$ of codimension $q$ on a $n$-dimen\-sio\-nal pseudo-Riemannian manifold 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 pse\-u\-do-Rieman\-ni\-an submersions and the fibres of different projections are orthogonal at common points. Relatively this metric the induced foliation $(G,\mathbb{F})$ on the graph is pseudo-Riemannian and the structure of the leaves of $(G,\mathbb{F})$ is described. Special attention is given to the structure of graphs of transversally (geodesically) complete pseudo-Riemannian foliations and totally geodesic pseudo-Riemannian ones.