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Article

Графики некоторого класса вполне геодезических слоений на псевдоримановых многообразиях

We investigate totally geodesic foliations (M, F) of arbitrary  codimemsion q on n-dimensional pseudo-Riemannian manifolds for which the induced metrics on leaves don't degenerate. We assume that the q-dimensional orthogonal distribution D to (M, F) is an Ehresmann connection for this foliation. Since the usual graph G(F) is not Hausdorff manifold in general, we investigate the graph G(F, D) of the foliation with an Ehresmann connection D introduced early by the author. This graph is always Hausdorff manifold. We prove that on the graph G(F, D) a pseudo-Riemannian metric is defined, with respect to which the induced foliation and the simple foliations formed by the fibers of the canonical projections are totally geodesic. It is proved that the leaves of the induced foliation on the graph are nondegenerately reducible pseudo-Riemannian manifolds and their structure is described. The application to parallel foliations on nondegenerately reducible pseudo-Riemannian manifolds is considered. It is shown that every foliation defined by the suspension of a homomorphism of the fundamental group of a pseudo-Riemannian manifold belongs to the investigated class of foliations.