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Regular version of the site

Article

Graphs of totally geodesic foliations on pseudo-Riemannian manifolds

Ufa Mathematical Journal. 2019. Vol. 11. No. 3. P. 30-44.

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 nondegenerate.
We assume that the q-dimensional orthogonal distribution 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 GM(F) of the foliation with an Ehresmann
connection M introduced early by the author. This graph is always a Hausdorff manifold.
We prove that on the graph GM(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.