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Spin(9) and almost complex structures on 16-dimensional manifolds

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Abstract

For a Spin(9)-structure on a Riemannian manifold M 16 we write explicitly the matrix ψ of its Kähler 2-forms and the canonical 8-form ΦSpin(9). We then prove that ΦSpin(9) coincides up to a constant with the fourth coefficient of the characteristic polynomial of ψ. This is inspired by lower dimensional situations, related to Hopf fibrations and to Spin(7). As applications, formulas are deduced for Pontrjagin classes and integrals of ΦSpin(9) and \({\Phi_{\rm Spin(9)}^2}\) in the special case of holonomy Spin(9).

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References

  1. Abe K.: Closed regular curves and the fundamental form on the projective spaces. Proc. Japan Acad. 68, 123–125 (1992)

    Article  MATH  Google Scholar 

  2. Abe K., Matsubara M.: Invariant forms on the exceptional symmetric Spaces FII and EIII. Proc. Korea Japan Conf. Transformation Group Theory(3), 1–15 (1997)

    Google Scholar 

  3. K. Abe, Matsubara, M. Erratum to [2], Private communication by K. Abe (2009)

  4. Agricola I.: The SRNÍ Lectures on non-integrable Geometries with Torsion. Arch. Math. (Brno) 42(Supplement), 5–84 (2006)

    MathSciNet  MATH  Google Scholar 

  5. Alekseevskij D.V.: Riemannian Spaces with exceptional Holonomy Groups. Funct. Anal. Prilozhen. 2, 97–105 (1968)

    Article  Google Scholar 

  6. Alekseevskij D.V., Marchiafava S.: Quaternionic structures on a manifold and subordinated structures. Ann. Mat. Pura Appl. 171, 205–273 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  7. Baez J.C.: The octonions. Bull. Amer. Math. Soc. 39, 145–205 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  8. Baez J.C.: The octonions. Bull. Amer. Math. Soc. 42, 213 (2005)

    Article  MathSciNet  Google Scholar 

  9. Berger M.: Du côté de chez Pu. Ann. Sci École Norm. Sup. 5, 1–44 (1972)

    MATH  Google Scholar 

  10. Besse A.: Einstein manifolds. Springer-Verlag, New York (1987)

    MATH  Google Scholar 

  11. Borel A., Hirzebruch F.: Characteristic classes and homogeneous spaces, I. Amer. J. Math. 80, 458–538 (1958)

    Article  MathSciNet  Google Scholar 

  12. Brown R.B., Gray A.: Riemannian manifolds with holonomy Spin(9), pp. 41–59. Differential Geometry (in honour of K. Yano Kinokuniya), Tokyo (1972)

    Google Scholar 

  13. Bryant R.L., Harvey R.: Submanifolds in hyper-Kähler gometry. J. Am. Math. Soc. 1, 1–31 (1989)

    MathSciNet  Google Scholar 

  14. Castrillon Lopez M., Gadea P.M., Mykytyuk I.V.: The canonical 8-form on Manifolds with Holonomy Group Spin(9). Int. J. Geom. Methods Mod. Phys. 7, 1159–1183 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Corlette K.: Archimedean superrigidity and hyperbolic geometry. Ann. Math. 135, 165–182 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  16. Dadok J., Harvey R., Morgan F.: Calibrations on \({{\mathbb R}^8}\). Trans. Amer. Math. Soc. 307, 1–40 (1988)

    MathSciNet  MATH  Google Scholar 

  17. Friedrich Th.: Weak Spin(9)-Structures on 16-dimensional Riemannian Manifolds. Asian J. Math. 5, 129–160 (2001)

    MathSciNet  MATH  Google Scholar 

  18. Friedrich Th.: Spin(9)-Structures and Connections with totally skew-symmetric Torsion. J. Geom. Phys. 47, 197–206 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gluck H., Warner F., Ziller W.: The geometry of the Hopf fibrations. Enseign. Math. 32, 173–198 (1986)

    MathSciNet  MATH  Google Scholar 

  20. Harvey R.: Spinors and calibrations. Academic Press, Boston (1990)

    MATH  Google Scholar 

  21. Harvey R., Lawson H.B. Jr.: Calibrated geometries. Acta Math. 148, 47–157 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  22. Joyce D.D.: Compact manifolds with special holonomy. Oxford University Press, Oxford (2000)

    MATH  Google Scholar 

  23. Joyce D.D.: Riemannian holonomy groups and calibrated geometry. Oxford University Press, Oxford (2007)

    MATH  Google Scholar 

  24. Kobayashi S., Nomizu K.: Foundations of differential geometry, vol. II. Interscience Publishers, New York (1969)

    MATH  Google Scholar 

  25. Parton, M., Piccinni, P.: Spheres with more than 7 vector fields: all the fault of Spin(9), arXiv:1107.0462, 1–12 (2011)

  26. Parton, M., Piccinni, P., Vuletescu, V.: 16-dimensional manifolds with a locally conformal parallel Spin(9) structure, in preparation

  27. Salamon, D.A., Walpuski, Th.: Notes on the Octonians, arXiv:1005.2820, 1–73 (2010)

  28. Salamon, S.M.: Riemannian geometry and holonomy groups. Longman Sci.Tech., (1989)

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Authors and Affiliations

  1. Dipartimento di Scienze, Università di Chieti-Pescara, viale Pindaro 87, 65127, Pescara, Italy

    Maurizio Parton

  2. Dipartimento di Matematica, Sapienza - Università di Roma, Piazzale Aldo Moro 2, 00185, Roma, Italy

    Paolo Piccinni

Authors
  1. Maurizio Parton
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  2. Paolo Piccinni
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Correspondence to Paolo Piccinni.

Additional information

M. Parton and P. Piccinni were supported by the MIUR under the PRIN Project “Geometria Differenziale e Analisi Globale”.

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Parton, M., Piccinni, P. Spin(9) and almost complex structures on 16-dimensional manifolds. Ann Glob Anal Geom 41, 321–345 (2012). https://doi.org/10.1007/s10455-011-9285-x

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  • DOI: https://doi.org/10.1007/s10455-011-9285-x

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