Skip to main content

Homogeneous almost normal Riemannian manifolds

Abstract

In this article, we introduce a newclass of compact homogeneous Riemannian manifolds (M = G/H, µ) almost normal with respect to a transitive Lie group G of isometries for which by definition there exists a G-left-invariant and an H-right-invariant inner product ν such that the canonical projection p: (G, ν) (G/H, µ) is a Riemannian submersion and the norm | · | of the product ν is at least the bi-invariant Chebyshev normon G defined by the space (M,µ).We prove the following results: Every homogeneous Riemannian manifold is almost normal homogeneous. Every homogeneous almost normal Riemannian manifold is naturally reductive and generalized normal homogeneous. For a homogeneous G-normal Riemannian manifold with simple Lie group G, the unit ball of the norm | · | is a Löwner-John ellipsoid with respect to the unit ball of the Chebyshev norm; an analogous assertion holds for the restrictions of these norms to a Cartan subgroup of the Lie group G. Some unsolved problems are posed.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    J. F. Adams, Lectures on Lie Groups (W. A Benjamin, Inc., New York-Amsterdam, 1969; Nauka, Moscow, 1979).

    MATH  Google Scholar 

  2. 2.

    K. M. Ball, “An elementary introduction to modern convex geometry,” Flavors of Geometry, Math. Sci. Res. Inst. Publ. 31, 1–58 (Cambridge Univ. Press, Cambridge, 1997).

    Google Scholar 

  3. 3.

    V. N. Berestovskiĭ and L. Guijarro, “A metric characterization of Riemannian submersions,” Ann. Global Anal. Geom. 18(6), 577–588 (2000).

    Article  MathSciNet  Google Scholar 

  4. 4.

    V.N. Berestovskiĭ, E. V. Nikitenko, and Yu. G. Nikonorov, “Classification of generalized normal homogeneous Riemannian manifolds of positive Euler characteristic,” Differential Geom. Appl. 29(4), 533–546 (2011).

    Article  MathSciNet  Google Scholar 

  5. 5.

    V. N. Berestovskiĭ and Yu. G. Nikonorov, “The Chebyshev norm on the Lie algebra of the motion group of a compact homogeneous Finsler manifold,” Sovrem. Mat. Prilozh. no. 60, Algebra, 98–121 (2008) [J. Math. Sci. (N. Y.) 161 (1), 97–121 (2009)].

    Google Scholar 

  6. 6.

    V. N. Berestovskiĭ and Yu. G. Nikonorov, “On δ-homogeneous Riemannian manifolds,” Differential Geom. Appl. 26(5), 514–535 (2008).

    Article  MathSciNet  Google Scholar 

  7. 7.

    V. N. Berestovskiĭ and Yu. G. Nikonorov, “On δ-homogeneous Riemannian manifolds. II,” Sibirsk. Mat. Zh. 50(2), 267–278 (2009) [SiberianMath. J., 50 (2), 214–222 (2009)].

    MATH  MathSciNet  Google Scholar 

  8. 8.

    V. N. Berestovskiĭ and Yu. G. Nikonorov, Generalized Normal Homogeneous Riemannian Metrics on Spheres and Projective Spaces, arXiv:1210:7727v1 [math.DG] 29 Oct 2012, 32 p.

    Google Scholar 

  9. 9.

    V. N. Berestovskiĭ and C. Plaut, “Homogeneous spaces of curvature bounded below,” J. Geom. Anal. 9(2), 203–219 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  10. 10.

    M. Berger, “Les variétés riemanniennes homogeènes normales simplement connexes à courbure strictement positive,” Ann. Scuola Norm. Sup. Pisa (3) 15(3), 179–246 (1961).

    MATH  MathSciNet  Google Scholar 

  11. 11.

    F. John, “Extremum problems with inequalities as subsidiary conditions,” Courant Anniversary Volume, 187–204 (Interscience Publishers, New York, 1948).

    Google Scholar 

  12. 12.

    B. Kostant, “On differential geometry and homogeneous spaces. I, II,” Proc. Nat. Acad. Sci. U.S.A. 42, 258–261, 354–357 (1956).

    Article  MathSciNet  Google Scholar 

  13. 13.

    B. Kostant “On holonomy and homogeneous spaces,” Nagoya Math. J. 12(1), 31–54 (1957).

    MATH  MathSciNet  Google Scholar 

  14. 14.

    O. Kowalski and L. Vanhekke, “Riemannian manifolds with homogeneous geodesics,” Boll. Un. Mat. Ital. B (7) 5(1), 189–246 (1991).

    MATH  MathSciNet  Google Scholar 

  15. 15.

    A. S. Lewis, “Group invariance and convex matrix analysis,” SIAM J. Matrix Anal. Appl. 17(4), 927–949 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  16. 16.

    K. Nomizu, “Invariant affine connections on homogeneous spaces,” Amer. J.Math. 76(1), 33–65 (1954).

    Article  MATH  MathSciNet  Google Scholar 

  17. 17.

    A. Selberg, “Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series,” J. IndianMath. Soc. (N.S.) 20, 47–87 (1956).

    MATH  MathSciNet  Google Scholar 

  18. 18.

    J. Szenthe, “Sur la connexion naturelle à torsion nulle,” Acta. Sci. Math. (Szeged) 38(3–4), 383–398 (1976).

    MATH  MathSciNet  Google Scholar 

  19. 19.

    É. B. Vinberg, “Invariant norms in compact Lie algebras,” Funkts. Anal. Prilozh. 2(2), 89–90 (1968) [Func. Anal. Appl. 2 (2), 177–179 (1968)].

    MATH  MathSciNet  Google Scholar 

  20. 20.

    O. S. Yakimova, “Weakly symmetric Riemannian manifolds with reductive isometry group,” Mat. Sb. 195(4), 143–160 (2004) [Sb.Math. 195 (3–4), 599–614 (2004)].

    Article  MathSciNet  Google Scholar 

  21. 21.

    W. Ziller, “Weakly symmetric spaces,” Progr. Nonlinear Differential Equations Appl. Topics in Geometry: in Memory of Joseph D’Atri. 20, 355–368 (Birkhäuser Boston, Boston, MA, 1996).

    MathSciNet  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to V. N. Berestovskiĭ.

Additional information

Original Russian Text © V.N. Berestovskiĭ, 2013, published in Matematicheskie Trudy, 2013, Vol. 16, No. 1, pp. 18–27.

About this article

Cite this article

Berestovskiĭ, V.N. Homogeneous almost normal Riemannian manifolds. Sib. Adv. Math. 24, 12–17 (2014). https://doi.org/10.3103/S1055134414010027

Download citation

Keywords

  • Weyl group
  • naturally reductive Riemannian manifold
  • Chebyshev norm
  • homogeneous normal Riemannian manifold
  • homogeneous generalized normal Riemannian manifold, homogeneous almost normal Riemannian manifold
  • Cartan subagebra
  • Löwner-John ellipsoid