Pseudo-Riemannian Foliations and Their Graphs

We prove that a foliation ( M , F ) of codimension q on a n -dimensional pseudo-Riemannian manifold with induced metrics on leaves 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...

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Published in:Lobachevskii journal of mathematics 2018, Vol.39 (1), p.54-64
Main Authors: Dolgonosova, A. Yu, Zhukova, N. I.
Format: Article
Language:English
Subjects:
Algebra
Analysis
Brownian motion
Geometry
Graphs
Mathematical Logic and Foundations
Mathematics
Mathematics and Statistics
Probability Theory and Stochastic Processes
Riemann manifold
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description We prove that a foliation ( M , F ) of codimension q on a n -dimensional pseudo-Riemannian manifold with induced metrics on leaves 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.
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subjects Algebra
Analysis
Brownian motion
Geometry
Graphs
Mathematical Logic and Foundations
Mathematics
Mathematics and Statistics
Probability Theory and Stochastic Processes
Riemann manifold
title Pseudo-Riemannian Foliations and Their Graphs
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