Transformation Groups

, Volume 18, Issue 3, pp 845–864 | Cite as

Spin(9) geometry of the octonionic Hopf fibration

  • Liviu Ornea
  • Maurizio Parton
  • Paolo Piccinni
  • Victor Vuletescu
Article

Abstract

We deal with Riemannian properties of the octonionic Hopf fibration S15S8, in terms of the structure given by its symmetry group Spin(9). In particular, we show that any vertical vector field has at least one zero, thus reproving the non-existence of S1 subfibrations. We then discuss Spin(9)-structures from a conformal viewpoint and determine the structure of compact locally conformally parallel Spin(9)-manifolds. Eventually, we give a list of examples of locally conformally parallel Spin(9)-manifolds.

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References

  1. [AF06]
    I. Agricola, T. Friedrich, Geometric structures of vectorial type, J. Geom. Phys. 56 (2006), no. 12, 2403–2414.MathSciNetMATHCrossRefGoogle Scholar
  2. [Ale68]
    Д. В. Алексеевский, Римановы пространства с необычными группами голономии, Функц. анализ и его прил. 2 (1968), вып. 2, 1–10. Engl. transl.: D. V. Alekseevskii, Riemannian spaces with exceptional holonomy groups, Func. Anal. Appl. 2 (1968), no. 2, 97–105.CrossRefGoogle Scholar
  3. [Bae02]
    J. C. Baez, The octonions, Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 2, 145–205.MathSciNetMATHCrossRefGoogle Scholar
  4. [Ber72]
    M. Berger, Du côté de chez Pu, Ann. Sci. École Norm. Sup. (4) 5 (1972), 1–44.MATHGoogle Scholar
  5. [BrGr72]
    R. B. Brown, A. Gray, Riemannian manifolds with holonomy group Spin(9), in: Differential Geometry, in Honor of K. Yano, Kinokuniya, Tokyo, 1972, pp. 41–59.Google Scholar
  6. [BG99]
    C. Boyer, K. Galicki, 3-Sasakian manifolds, in: Surveys in Differential Geometry: Essays on Einstein Manifolds, Int. Press, Boston, MA, 1999, pp. 123–184.Google Scholar
  7. [BGMR98]
    C. P. Boyer, K. Galicki, B. M. Mann, E. G. Rees, Compact 3-Sasakian 7-manifolds with arbitrary second Betti number, Invent. Math. 131 (1998), no. 2, 321–344.MathSciNetMATHCrossRefGoogle Scholar
  8. [Bru92]
    M. Bruni, Sulla parallelizzazione esplicita dei prodotti di sfere, Rend. di Mat., serie VII 12 (1992), 405–423.MathSciNetMATHGoogle Scholar
  9. [Cor92]
    K. Corlette, Archimedean superrigidity and hyperbolic geometry, Ann. Math. (2) 135 (1992), no. 1, 165–182.MathSciNetMATHCrossRefGoogle Scholar
  10. [Cox91].
    H. S. M. Coxeter, Regular Complex Polytopes, 2nd ed., Cambridge University Press, Cambridge, 1991.MATHGoogle Scholar
  11. [CP99].
    D. M. J. Calderbank, H. Pedersen, Einstein−Weyl geometry, in: Surveys in Differential Geometry: Essays on Einstein Manifolds, Int. Press, Boston, MA, 1999, pp. 387–423.Google Scholar
  12. [Dea08]
    O. Dearricott, n-Sasakian manifolds, Tohoku Math. J. (2) 60 (2008), no. 3, 329–347.MathSciNetMATHCrossRefGoogle Scholar
  13. [DO98]
    S. Dragomir, L. Ornea, Locally Conformal Kähler Geometry, Progress in Math., Vol. 155, Birkhäuser, Boston, MA, 1998.MATHCrossRefGoogle Scholar
  14. [DS01]
    R. De Sapio, On Spin(8) and triality: A topological approach, Expo. Math. 19 (2001), no. 2, 143–161.MathSciNetMATHCrossRefGoogle Scholar
  15. [FO08]
    J. Figueroa-O’Farrill, A geometric construction of the exceptional Lie algebras F 4 and E 8, Comm. Math. Phys. 283 (2008), no. 3, 663–674.MathSciNetMATHCrossRefGoogle Scholar
  16. [Fri01]
    T. Friedrich, Weak Spin(9)-structures on 16-dimensional Riemannian manifolds, Asian J. Math. 5 (2001), no. 1, 129–160.MathSciNetMATHGoogle Scholar
  17. [Gau95]
    P. Gauduchon, Structures de Weyl−Einstein, espaces de twisteurs et variétés de type S 1 × S 3, J. Reine Angew. Math. 469 (1995), 1–50.MathSciNetMATHGoogle Scholar
  18. [GOPP06]
    R. Gini, L. Ornea, M. Parton, P. Piccinni, Reduction of Vaisman structures in complex and quaternionic geometry, J. Geom. Phys. 56 (2006), no. 12, 2501–2522.MathSciNetMATHCrossRefGoogle Scholar
  19. [GWZ86]
    H. Gluck, F. Warner, W. Ziller, The geometry of the Hopf fibrations, Enseign. Math. (2) 32 (1986), no. 3–4, 173–198.MathSciNetMATHGoogle Scholar
  20. [Har90]
    F. R. Harvey, Spinors and Calibrations, Perspectives in Mathematics, Vol. 9, Academic Press, Boston, MA, 1990.MATHGoogle Scholar
  21. [Hus94]
    D. Husemoller, Fibre Bundles, 3rd ed., Graduate Texts in Mathematics, Vol. 20, Springer-Verlag, New York, 1994.CrossRefGoogle Scholar
  22. [IPP06]
    S. Ivanov, M. Parton, P. Piccinni, Locally conformal parallel G 2 and Spin(7) manifolds, Math. Res. Lett. 13 (2006), no. 2–3, 167–177.MathSciNetMATHGoogle Scholar
  23. [Kol02]
    A. Kollross, A classification of hyperpolar and cohomogeneity one actions, Trans. Amer. Math. Soc. 354 (2002), no. 2, 571–612.MathSciNetMATHCrossRefGoogle Scholar
  24. [LV92]
    B. Loo, A. Verjovsky, The Hopf fibration over S 8 admits no S 1-subfibration, Topology 31 (1992), no. 2, 239–254.MathSciNetMATHCrossRefGoogle Scholar
  25. [Mur89]
    S. Murakami, Exceptional simple Lie groups and related topics in recent differential geometry, in: Differential Geometry and Topology (Tianjin 1986–87), Lecture Notes in Mathematics, Vol. 1369, Springer, Berlin, 1989, pp. 183–221.CrossRefGoogle Scholar
  26. [O’N83]
    B. O’Neill, Semi-Riemannian Geometry with Applications to Relativity, Pure and Applied Mathematics, Vol. 103, Academic Press, New York, 1983.MATHGoogle Scholar
  27. [OP97]
    L. Ornea, P. Piccinni, Locally conformal Kähler structures in quaternionic geometry, Trans. Am. Math. Soc. 349 (1997), no. 2, 641–655.MathSciNetMATHCrossRefGoogle Scholar
  28. [OV03]
    L. Ornea, M. Verbitsky, Structure theorem for compact Vaisman manifolds, Math. Res. Lett. 10 (2003), no. 5–6, 799–805.MathSciNetMATHGoogle Scholar
  29. [Par01a]
    M. Parton, Hermitian and special structures on products of spheres, PhD thesis, Dipartimento di Matematica, Università di Pisa, 2001, www.sci.unich.it/~parton/rice/main.pdf.
  30. [Par01b]
    M. Parton, Old and new structures on products of spheres, in: Global Differential Geometry: the Mathematical Legacy of Alfred Gray, Bilbao, 2000, American Math. Soc., Providence, RI, 2001, pp. 406–410.CrossRefGoogle Scholar
  31. [Par03]
    M. Parton, Explicit parallelizations on products of spheres and Calabi−Eckmann structures, Rend. Ist. Mat. Univ. Trieste 35 (2003), no. 1–2, 61–67.MathSciNetMATHGoogle Scholar
  32. [PP12]
    M. Parton, P. Piccinni, Spin(9) and almost complex structures on 16-dimensional manifolds, Ann. Global Anal. Geom. 41 (2012), no. 3, 321–345.MathSciNetMATHCrossRefGoogle Scholar
  33. [PP13]
    M. Parton, P. Piccinni, Spheres with more than 7 vector fields: All the fault of Spin (9), Linear Algebra Appl. 438 (2013), no. 3, 1113–1131.MathSciNetMATHCrossRefGoogle Scholar
  34. [Ste99]
    N. Steenrod, The Topology of Fibre Bundles, 1st paperback ed., Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1999.MATHGoogle Scholar
  35. [Var01]
    V. S. Varadarajan, Spin(7)-subgroups of SO(8) and Spin(8), Expo. Math. 19 (2001), no. 2, 163–177.MathSciNetMATHCrossRefGoogle Scholar
  36. [Zil82]
    W. Ziller, Homogeneous Einstein metrics on spheres and projective spaces, Math. Ann. 259 (1982), no. 3, 351–358.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Liviu Ornea
    • 1
    • 2
  • Maurizio Parton
    • 3
  • Paolo Piccinni
    • 4
  • Victor Vuletescu
    • 1
  1. 1.University of Bucharest, Faculty of Mathematics and InformaticsBucharestRomania
  2. 2.Institute of Mathematics “Simion Stoilow” of the Romanian AcademyBucharestRomania
  3. 3.Università di Chieti-Pescara, Dipartimento di EconomiaPescaraItaly
  4. 4.Dipartimento di MatematicaSapienza—Università di RomaRomaItaly

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