Pseudo-Riemannian foliations and their graphs

Lobachevskii Journal of Mathematics. 2018. Vol. 39. No. 1. P. 54–64.

A. Yu. Dolgonosova, Жукова Н. И.

We prove that a foliation (M;F) of codimension q on a ndimensional 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 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.

Ключевые слова: Pseudo-Riemannian foliation, geodesically invariant distribution, graph of a foliation, Ehresmann connection of a foliation

ПУБЛИКАЦИЯ ПОДГОТОВЛЕНА ПО РЕЗУЛЬТАТАМ ПРОЕКТА: Топология и хаос в динамике систем, слоений и деформации алгебр Ли (2018)