Lobachevskii Journal of Mathematics

, Volume 39, Issue 1, pp 54–64 | Cite as

Pseudo-Riemannian Foliations and Their Graphs

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Abstract

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.

Keywords and phrases

Pseudo-Riemannian foliation graph of a foliation geodesically invariant distribution Ehresmann connection of a foliation 

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References

  1. 1.
    B. O’Neill, Semi-Riemannian Geometry with Applications to Relativity (Academic, New York, London, 1983).MATHGoogle Scholar
  2. 2.
    B. Reinhart, “Foliated manifolds with bundle-like metrics,” Ann. Math. 69, 119–132 (1958).MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    P. Molino, Riemannian Foliations, Vol. 73 of Progress in Mathematics (Birkhauser, Boston, 1988).CrossRefGoogle Scholar
  4. 4.
    A. D. Lewis, “Affine connections and distributions,” Rep. Math. Phys. 42, 135–164 (1998).MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    N. I. Zhukova and A. Y. Dolgonosova, “The automorphism groups of foliations with transverse linear connection,” Cent. Eur. J. Math. 11, 2076–2088 (2013).MathSciNetMATHGoogle Scholar
  6. 6.
    H. E. Winkelnkemper, “The graph of a foliation,” Ann. Glob. Anal. Geom. 1 (3), 51–75 (1983).MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    H. Wu, “On the de Rham decomposition theorem,” Illinois J.Math. 8, 291–311 (1964).MathSciNetMATHGoogle Scholar
  8. 8.
    R. A. Blumenthal and J. J. Hebda, “Ehresmann connections for foliations,” Indiana Univ. Math. J. 33, 597–611 (1984).MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    N. I. Zhukova, “The graph of a foliation with Ehresmann connection and stability of leaves,” Russ. Math. 38, 76–79 (1994).MathSciNetMATHGoogle Scholar
  10. 10.
    N. I. Zhukova, “Local and global stability of compact leaves and foliations,” J. Math. Phys., Anal. Geom. 9, 400–420 (2013).MathSciNetMATHGoogle Scholar
  11. 11.
    N. I. Zhukova, “Singular foliations with Ehresmann connections and their holonomy groupoids,” Banach Center Publ. 76, 471–490 (2007).MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    K. Yokumoto, “Mutual exclusiveness along spacelike, timelike, and lightlike leaves in totally geodesic foliations of lightlike complete Lorentzian two-dimensional tori,” HokkaidoMath. J. 31, 643–663 (2000).MathSciNetCrossRefGoogle Scholar
  13. 13.
    C. Boubel, P. Mounoud, and C. Tarquini, “Lorentzian foliations on 3-manifolds,” Ergodic Theory Dynam. System 26, 1339–1362 (2006).MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    E. Ghys, “Deformations de flots d’Anosov et de groupes fuchsiens,” Ann. Inst. Fourier 42, 209–247 (1992).MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    N. I. Zhukova and E. A. Rogozhina, “Classification of compact Lorentzian 2-orbifolds with non-compact full isometry groups,” Sib. Math. J. 53, 1037–1050 (2012).MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    D. Bump, Group Representation Theory. http://sporadic.stanford.edu/bump/group/. Accessed 2010.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Department of Informatics, Mathematics and Computer SciencesNational Research University Higher School of EconomicsMoscowRussia

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