Advertisement

On Compact Affine Quaternionic Curves and Surfaces

  • Graziano GentiliEmail author
  • Anna Gori
  • Giulia Sarfatti
Article
  • 31 Downloads

Abstract

This paper is devoted to the study of affine quaternionic manifolds and to a possible classification of all compact affine quaternionic curves and surfaces. It is established that on an affine quaternionic manifold there is one and only one affine quaternionic structure. A direct result, based on the celebrated Kodaira Theorem that studies compact complex manifolds in complex dimension 2, states that the only compact affine quaternionic curves are the quaternionic tori and the primary Hopf surface \(S^3\times S^1\). As for compact affine quaternionic surfaces, we restrict to the complete ones: the study of their fundamental groups, together with the inspection of all nilpotent hypercomplex simply connected 8-dimensional Lie Groups, identifies a path towards their classification.

Keywords

Affine quaternionic manifolds Fundamental groups of compact affine quaternionic surfaces 

Mathematics Subject Classification

30G35 53C15 

Notes

References

  1. 1.
    Aslaksen, H.: Quaternionic determinants. Math. Intell. 18, 57–65 (1996)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Angella, D., Bisi, C.: Slice-quaternionic Hopf surfaces. J. Geom. Anal. 29, 1837–1858 (2019)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Auslander, L.: Four dimensional compact locally hermitian manifolds. Trans. Am. Math. Soc. 84, 379–391 (1957)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bisi, C., Gentili, G.: On quaternionic tori and their moduli space. J. Noncommut. Geom. 12, 473–510 (2018)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Conway, J.H., Smith, D.A.: On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry. A.K.Peters, Natick, MA (2003)CrossRefGoogle Scholar
  6. 6.
    De Leo, S., Scolarici, G., Solombrino, L.: Quaternionic eigenvalue problem. J. Math. Phys. 43, 5815–5829 (2002)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Dotti, I.G., Fino, A.: Hypercomplex eight-dimensional nilpotent Lie groups. J. Pure Appl. Algebra 184, 41–57 (2003)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Dotti, I.G., Fino, A.: Abelian hypercomplex eight-dimensional nilmanifolds. Ann. Glob. Anal. Geom. 18, 47–59 (2000)CrossRefGoogle Scholar
  9. 9.
    Dotti, I.G., Fino, A.: Hypercomplex nilpotent Lie groups. Contemp. Math. 288, 310–314 (2001)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Fillmore, J.P., Scheuneman, J.: Fundamental groups of compact complete locally affine complex surfaces. Pac. J. Math. 44, 487–496 (1973)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Fried, D., Goldman, W., Hirsh, M.: Affine manifolds with nilpotent holonomy. Comment. Math. Helvetici 56, 487–523 (1981)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Gentili, G., Gori, A., Sarfatti, G.: A direct approach to quaternionic manifolds. Math. Nachr. 290, 321–331 (2017)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Gentili, G., Gori, A., Sarfatti, G.: Quaternionic toric manifolds. J. Symplectic Geom. 17, 267–300 (2019)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Gentili, G., Stoppato, C., Struppa, D.C.: Regular Functions of a Quaternionic Variable. Springer Monographs in Mathematics. Springer, Berlin-Heidelberg (2013)CrossRefGoogle Scholar
  15. 15.
    Ghiloni R., Perotti A.: Slice regular functions of several Clifford variables in 9th International Conference on Mathematical Problems in Engineering, Aerospace and Sciences : ICNPAA 2012 : Vienna, AustriaGoogle Scholar
  16. 16.
    Gordon, C.S., Wilson, E.N.: The spectrum of the Laplacian on Riemannian Heisenberg manifolds. Mich. Math. J. 33, 253–271 (1986)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Inoue, M., Kobayashi, S., Ochiai, T.: Holomorphic affine connections on compact complex surfaces. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27, 247–264 (1980)Google Scholar
  18. 18.
    Kaplan, A.: Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms. Trans. Am. Math. Soc. 258, 147–153 (1980)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Kobayashi, S., Horst, C.: Topics in Complex Differential Geometry, Complex Differential Geometry, 4-66, DMV Sem., 3, Birkhaüser, Basel (1983)Google Scholar
  20. 20.
    Kodaira, K.: On the structure of compact complex analytic surfaces I. Am. J. Math. 86, 751–798 (1964)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Kodaira, K.: On the structure of compact complex analytic surfaces II. Am. J. Math. 88, 682–721 (1966)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Kodaira, K.: On the structure of compact complex analytic surfaces III. Am. J. Math. 90, 55–83 (1968)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Rodman, L.: Topics in Quaternion Linear Algebra. Princeton Series in Applied Mathematics. Princeton University Press, Princeton, NJ, . xii+363 pp (2014)Google Scholar
  24. 24.
    Malcev, A. I.: On a class of homogeneous spaces. Amer. Math. Soc. Translation, p. 33 (1951)Google Scholar
  25. 25.
    Matsushima, Y.: Affine structures on complex manifolds Osaka. J. Math. 5, 215–222 (1968)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Obata, M.: Affine connections on manifolds with almost complex, quaternion or Hermitian structure Jap. J. Math. 26, 43–77 (1956)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Salamon, S.M.: Differential geometry of quaternionic manifolds. Ann. Sci. École Norm. Sup. 9(4), 31–55 (1986)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Sommese, A.: Quaternionic manifolds. Math. Ann. 212, 191–214 (1974/75)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Vitter, A.: Affine structures on compact complex manifolds. Invent. Math. 17, 231–244 (1972)MathSciNetCrossRefGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2019

Authors and Affiliations

  1. 1.Dipartimento di Matematica e Informatica “U. Dini”Università di FirenzeFirenzeItaly
  2. 2.Dipartimento di MatematicaUniversità di MilanoMilanoItaly
  3. 3.Dipartimento di Ingegneria Industriale e Scienze MatematicheUniversità Politecnica delle MarcheAnconaItaly

Personalised recommendations