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Totally geodesic hypersurfaces of homogeneous spaces

Abstract

We show that a simply connected Riemannian homogeneous space M which admits a totally geodesic hypersurface F is isometric to either (a) the Riemannian product of a space of constant curvature and a homogeneous space, or (b) the warped product of the Euclidean space and a homogeneous space, or (c) the twisted product of the line and a homogeneous space (with the warping/twisting function given explicitly). In the first case, F is also a Riemannian product; in the last two cases, it is a leaf of a totally geodesic homogeneous fibration. Case (c) can alternatively be characterized by the fact that M admits a Riemannian submersion onto the universal cover of the group SL(2) equipped with a particular left-invariant metric, and F is the preimage of the two-dimensional solvable totally geodesic subgroup.

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References

  1. D. Alekseevsky and Y. Nikonorov, Compact Riemannian manifolds with homogeneous geodesics, SIGMA Symmetry Integrability Geometry: Methods and Applications 5 (2009), Paper 093.

  2. A. Besse, Einstein Manifolds, Springer, Berlin, Heidelberg, New York, 1987.

    MATH  Book  Google Scholar 

  3. R. Blumenthal and J. Hebda, de Rham decomposition theorems for foliated manifolds, Annales de l’Institut Fourier (Grenoble) 33 (1983), 183–198.

    MATH  MathSciNet  Article  Google Scholar 

  4. G. Cairns, A. Hinić Galić and Y. Nikolayevsky, Totally geodesic subalgebras of nilpotent Lie algebras, Journal of Lie Theory 23 (2013), 1023–1049.

    MATH  MathSciNet  Google Scholar 

  5. G. Cairns, A. Hinić Galić and Y. Nikolayevsky, Totally geodesic subalgebras of filiform nilpotent Lie algebras, Journal of Lie Theory 23 (2013), 1051–1074.

    MATH  MathSciNet  Google Scholar 

  6. B.Y. Chen and T. Nagano, Totally geodesic submanifolds of symmetric spaces. II, Duke Mathematical Journal 45 (1978), 405–425.

    MATH  MathSciNet  Article  Google Scholar 

  7. É. Cartan, Sur une classe remarquable d’espaces de Riemann. II, Bulletin de la Société Mathématique de France 55 (1927), 114–134.

    MathSciNet  Google Scholar 

  8. Z. Dušek, The existence of homogeneous geodesics in homogeneous pseudo-Riemannian and affine manifolds, Journal of Geometry and Physics 60 (2010), 687–689.

    MATH  MathSciNet  Article  Google Scholar 

  9. P. Eberlein, Geometry of 2-step nilpotent groups with a left invariant metric. II, Transactions of the American Mathematical Society 343 (1994), 805–828.

    MATH  MathSciNet  Google Scholar 

  10. C. Gordon, Homogeneous Riemannian manifolds whose geodesics are orbits, in Topics in Geometry: in Memory of Joseph D’Atri, Progress in Nonlinear Differential Equations and Their Applications, Vol. 20, Birkhäuuser, Boston, MA, 1996, pp. 155–174.

    Chapter  Google Scholar 

  11. S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Pure and Applied Mathematics, Vol. 80, Academic Press, New York-London, 1978.

    MATH  Google Scholar 

  12. K. Hofmann, Hyperplane subalgebras of real Lie algebras, Geometriae Dedicata 36 (1990), 207–224.

    MATH  MathSciNet  Article  Google Scholar 

  13. T. Jentsch, A. Moroianu and U. Semmelmann, Extrinsic hyperspheres in manifolds with special holonomy, Differential Geometry and its Applications 31 (2013), 104–111.

    MATH  MathSciNet  Article  Google Scholar 

  14. V. Kaĭzer, Conjugate points of left-invariant metrics on Lie groups, (Russian) Izv. Vyssh. Uchebn. Zaved. Mat. 1990, no. 11, 27–37; translation in Soviet Math. (Iz. VUZ) 34 (1990), 32–44.

  15. M. Kerr and T. Payne, The geometry of filiform nilpotent Lie groups, Rocky Mountain Journal of Mathematics 40 (2010), 1587–1610.

    MATH  MathSciNet  Article  Google Scholar 

  16. O. Kowalski and J. Szenthe, On the existence of homogeneous geodesics in homogeneous Riemannian manifolds, Geometriae Dedicata 81 (2000), 209–214.

    MATH  MathSciNet  Article  Google Scholar 

  17. S. Lie, Theorie der Transformationsgruppen, Mathematische Annalen 16 (1880), 441–528.

    MATH  MathSciNet  Article  Google Scholar 

  18. J. Tits, Sur une classe de groupes de Lie résolubles, Bulletin of the Belgian Mathematical Society 11 (1959), 100–115.

    MATH  MathSciNet  Google Scholar 

  19. K. Tojo, Totally geodesic submanifolds of naturally reductive homogeneous spaces, Tsukuba Journal of Mathematics 20 (1996), 181–190.

    MATH  MathSciNet  Google Scholar 

  20. K. Tojo, Normal homogeneous spaces admitting totally geodesic hypersurfaces, Journal of the Mathematical Society of Japan 49 (1997), 781–815.

    MATH  MathSciNet  Article  Google Scholar 

  21. K. Tsukada, Totally geodesic hypersurfaces of naturally reductive homogeneous spaces, Osaka Journal of Mathematics 33 (1996), 697–707.

    MATH  MathSciNet  Google Scholar 

  22. K. Tsukada, Totally geodesic submanifolds of Riemannian manifolds and curvature-invariant subspaces, Kodai Mathematical Journal 19 (1996), 395–437.

    MATH  MathSciNet  Article  Google Scholar 

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Correspondence to Y. Nikolayevsky.

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Supported by ARC Discovery grant DP130103485

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Nikolayevsky, Y. Totally geodesic hypersurfaces of homogeneous spaces. Isr. J. Math. 207, 361–375 (2015). https://doi.org/10.1007/s11856-015-1158-8

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  • DOI: https://doi.org/10.1007/s11856-015-1158-8

Keywords

  • Homogeneous Space
  • Warped Product
  • Riemannian Submersion
  • Geodesic Submanifolds
  • Riemannian Product