Transformation Groups

, Volume 22, Issue 4, pp 1143–1183 | Cite as


  • SHAOQIANG DENGEmail author


In this paper, we study normal homogeneous Finsler spaces. We first define the notion of a normal homogeneous Finsler space using the method of isometric submersion of Finsler metrics. Then we study the geometric properties. In particular, we establish a technique to reduce the classification of normal homogeneous Finsler spaces of positive flag curvature to an algebraic problem. The main result of this paper is a classification of positively curved normal homogeneous Finsler spaces. It turns out that a coset space G/H admits a positively curved normal homogeneous Finsler metric if and only if it admits a positively curved normal homogeneous Riemannian metric. We will also give a complete description of the coset spaces admitting non-Riemannian positively curved normal homogeneous Finsler spaces.


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  1. 1.
    S. Aloff, N. Wallach, An infinite family of distinct 7-manifolds admittiing positively curved Riemannian structures, Bull. Amer. Math. Soc. 81 (1975), 93–97.CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    J. C. Álvarez Paiva, C. E. Durán, Isometric submersion of Finsler manifolds, Proc. Amer. Math. Soc. 129 (2001), 2409–2417.CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Я. B. Бaзaйкин, Oб oднoм ceмeйcтвe 13-мepныx зaмкнутыx pимaнoвыx мнoгooбpaзий пoлoжитeльнoй кpивизны, Cиб. Maтeм. жуpн. 37 (1996), no. 6, 1219–1237. Engl. transl.: Ya. V. Bazaikin, On a certain family of closed 13-dimensional Riemannian manifolds of positive curvature, Sib. Math. J. 37 (1996), no. 6, 1068–1085.Google Scholar
  4. 4.
    L. Bérard-Bergery, Les variétes Riemannienes homogénes simplement connexes de dimension impair à courbure strictement positive, J. Math. Pure Appl. 55 (1976), 47–68.zbMATHGoogle Scholar
  5. 5.
    D. Bao, S. S. Chern, Z. Shen, An Introduction to Riemann-Finsler Geometry, Springer- Verlag, New York, 2000.Google Scholar
  6. 6.
    M. Berger, Les variétés riemanniennes homogénes normales simplement connexes à courbure strictement positive, Ann. Scuola Norm. Sup. Pisa (3) 15 (1961), 179–246.Google Scholar
  7. 7.
    A. Besse, Einstein Manifolds, Springer-Verlag, Berlin, 1987. Russian transl.: A. Бece, Mнoгooбpaзия Эйнштeйнa, Mиp, M., 1990.Google Scholar
  8. 8.
    V. N. Berestovskii, Yu. G. Nikonorov, On δ-homogeneous Riemannian manifolds, Differ. Geom. Appl. 26 (2008), 514–535.CrossRefzbMATHGoogle Scholar
  9. 9.
    A. Borel, Some remarks about Lie groups transitive on spheres and tori, Bull. AMS 55 (1940), 580–587.CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    S. S. Chern, Z. Shen, Riemann-Finsler Geometry, Nankai Tracts in Mathematics, Vol. 6, World Scientific, Hackensack, NJ, 2005.Google Scholar
  11. 11.
    S. Deng, Fixed points of isometries of a Finsler space, Publ. Math. Debrecen 72(2008), 469–474.zbMATHMathSciNetGoogle Scholar
  12. 12.
    S. Deng, Finsler metrics and the degree of symmetry of a closed manifold, Indiana U. Math. J. 60 (2011), 713–727.CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    S. Deng, Homogeneous Finsler Spaces, Springer, New York, 2012.CrossRefzbMATHGoogle Scholar
  14. 14.
    S. Deng, M. Xu, (α1 ; α2)-metrics and Clifford-Wolf homogeneity, J. Geom. Anal. 26 (2016), 2282–2321.Google Scholar
  15. 15.
    O. Dearricott, A 7-dimensional manifold with positive curvature, Duke Math. J. 158 (2011), 307–346.CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    S. Deng, Z. Hou, Invariant Finsler metrics on homogeneous manifolds, J. Phys. A: Math. Gen. 37 (2004), 8245–8253.CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    S. Deng, Z. Hu, Curvatures of homogeneous Randers spaces, Adv. Math. 240(2013), 194–226.CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    J. Eschenburg, New examples of manifolds of positive curvature, Invent. Math. 66 (1982), 469–480.CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    K. Grove, B. Wilking, W. Ziller, Positively curved cohomogeneity one manifolds and 3-Sasaki geometry, J. Diff. Geom. 78 (2008), 33–111.CrossRefzbMATHGoogle Scholar
  20. 20.
    K. Grove, L. Verdiani, W. Ziller, An exotic T 1 S 4 with positive curvature, Geom. Funct. Anal. 21 (2011), 499–524.CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Z. Hu, S. Deng, Homogeneous Randers spaces with positive flag curvature and isotropic S-curvature, Math. Z. 270 (2012), 989–1009.CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, New York, 1978.Google Scholar
  23. 23.
    L. Huang, On the fundamental equations of homogeneous Finsler spaces, Differ. Geom. Appl. 40 (2015), 187–208.CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, Interscience Publishers, Wiley, New York, Vol. 1, 1963, Vol. 2, 1969. Russian transl.: Ш. Кoбaяcи, К. Hoмидзу, Ocнoвы диффepeнциaльнoй гeoмeтpии, тт. 1 и 2 Haукa. M., 1981.Google Scholar
  25. 25.
    L. Kozma, Weinstein's theorem for Finsler spaces, Kyoto J. Math. 46 (2006), 377–382.CrossRefzbMATHGoogle Scholar
  26. 26.
    D. Montgomery, H. Samelson, Transformation groups of spheres, Ann. Math. 44 (1943), 454–470.CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    A. Л. Oнишник, O тpaнзитивныe кoмпaктныx гpуппax пpeoбpaзoвaний, Maтeм. cб. 60(102) (1963), no. 4, 447–485. Engl. transl.: A. L. Onisĉik, Transitive compact transformation groups, Amer. Math. Soc. Trans. 55 (2) (1966), 153–194.Google Scholar
  28. 28.
    G. Randers, On an assymmetric metric in the four-space of general relativity, Physics Rev. 59 (1941), 195–199.CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Z. Shen, Volume comparison and its applications in Riemann-Finsler geometry, Adv. Math. 128 (1997), 306–328.CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Z. Shen, Lectures on Finsler Geometry, World Scientific, Singapore, 2001.Google Scholar
  31. 31.
    L. Verdianni, W. Ziller, Positively curved homogeneous metrics on spheres, Math. Z. 261 (2009), no. 3, 473–488.CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    N. R. Wallach, Compact homogeneous Riemannian manifolds with strictly positive curvature, Ann. Math. 96 (1972), 277–295.CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    M. Wang, W. Ziller, On normal homogeneous Einstein manifolds, Ann. Sci. I'ENS 18 (1985), 563–633.zbMATHMathSciNetGoogle Scholar
  34. 34.
    B. Wilking, The normal homogeneous space (SU(3) × SO(3))/U*(2) has positive sectional curvature, Proc. of the AMS 127 (1999), 1191–1194.CrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    B. Wilking, W. Ziller, Revisiting homogeneous spaces with positive curvature, J. Reine Angew. Math., to appear, arXiv:1503.06256 (2015).Google Scholar
  36. 36.
    M. Xu, S. Deng, L. Huang, Z. Hu, Even-dimensional homogeneous Finsler spaces with positive flag curvature, Indiana U. Math. J., to appear, arXiv:1407.3582v2 (2015).Google Scholar
  37. 37.
    M. Xu, J. A.Wolf, Sp(2)/U(1) and a positive curvature problem, Differ. Geom. Appl. 42 (2015), 115–124.CrossRefzbMATHMathSciNetGoogle Scholar
  38. 38.
    W. Ziller, Weakly symmetric spaces, in: Topics in Geometry, in the memory of Joseph D’Atric, Progr. Nonlinear Diff. Eqs. Appl., Vol. 20, Birkhäuser Boston, Boston, MA, 1996, pp. 355–368.Google Scholar

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© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.College of MathematicsTianjin Normal UniversityTianjinChina
  2. 2.School of Mathematical Sciences and LPMCNankai UniversityTianjinChina

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