On Invariants of Almost Symplectic Connections

  • R. Albuquerque
  • R. Picken


We study the irreducible decomposition under \(Sp(2n,{\mathbb R})\) of the space of torsion tensors of almost symplectic connections. Then a description of all symplectic quadratic invariants of torsion-like tensors is given. When applied to a manifold M with an almost symplectic structure, these instruments give preliminary insight for finding a preferred linear almost symplectic connection on M. We rediscover Ph. Tondeur’s Theorem on almost symplectic connections. Properties of torsion of the vectorial kind are deduced.


Almost symplectic Linear connection Torsion Quadratic invariant. 

Mathematics Subject Classification (2010)

53A55 53B05 53D15 


  1. 1.
    Agricola, I., Thier, Chr.: The geodesics of metric connections with vectorial torsion. Ann. Glob. Anal. Geom. 26, 321–332 (2004)CrossRefMathSciNetMATHGoogle Scholar
  2. 2.
    Albuquerque, R., Rawnsley, J.: Twistor theory of symplectic manifolds. J. Geom. Phys. 56, 214–246 (2006)CrossRefADSMathSciNetMATHGoogle Scholar
  3. 3.
    Bourgeois, F., Cahen, M.: A variational principle for symplectic connections. J. Geom. Phys. 30(3), 233–265 (1999)CrossRefADSMathSciNetMATHGoogle Scholar
  4. 4.
    Burstall, F., Rawnsley, J.: Affine connections with W=0. arXiv:math/0702032 (2007)
  5. 5.
    Cahen, M., Gutt, S., Horowitz, J., Rawnsley, J.: Homogeneous symplectic manifolds with ricci-type curvature. J. Geom. Phys. 38(2), 140–151 (2001)CrossRefADSMathSciNetMATHGoogle Scholar
  6. 6.
    Cahen, M., Gutt, S., Horowitz, J., Rawnsley, J.: Moduli space of symplectic connections of ricci type on t 2n; a formal approach. J. Geom. Phys. 46, 174–192 (2003)CrossRefADSMathSciNetMATHGoogle Scholar
  7. 7.
    Cahen, M., Gutt, S., Rawnsley, J.: Symmetric symplectic spaces with Ricci-type curvature, Kluwer Acad., Math. Phys. Stud., 22, 2000, Conférence Moshé Flato 1999, Vol. II (Dijon)Google Scholar
  8. 8.
    Gelfand, I., Retahk, V., Shubin, M.: Fedosov manifolds. Adv. Math. 136, 104–140 (1998)CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    Habermann, K., Habermann, L., Rosenthal, P.: Symplectic Yang–Mills theory, Ricci tensor, and connections. Calc. Variations Part. Dif. Eq. 30(2) (2007)Google Scholar
  10. 10.
    Kraft, H., Procesi, C.: Classical Invariant Theory A Primer, Preliminary version (1996)Google Scholar
  11. 11.
    Nannicini, A.: Twistor methods in conformal almost symplectic geometry. Rend. Istit. Mat. Univ. Trieste 34, 215–234 (2002)MathSciNetMATHGoogle Scholar
  12. 12.
    Rangarajan, G., Neri, F.: Canonical representations of \(sp(2n, {\mathbb R})\). J. Math. Phys. 33(4) (1992)Google Scholar
  13. 13.
    Tondeur, P.: Affine Zusammenhänge auf Mannigfaltigkeiten mit fastsymplektischer Struktur. Comment. Math. Helv. 36, 234–244 (1962)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Urakawa, H.: Yang–Mills theory over compact symplectic manifolds. Ann. Global Anal. Geom. 25(4), 365–402 (2004)CrossRefMathSciNetMATHGoogle Scholar
  15. 15.
    Vaisman, I.: Symplectic curvature tensors. Monatsh. Math. 100(4), 299–327 (1985)CrossRefMathSciNetMATHGoogle Scholar
  16. 16.
    Vaisman, I.: Symplectic twistor spaces. J. Geom. Phys. 3(4), 507–524 (1986)CrossRefADSMathSciNetMATHGoogle Scholar
  17. 17.
    Weyl, H.: Classical Groups, their Invariants and Representations. Princeton University Press (1946)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Departamento de Matemática da Universidade de Évora and Centro de Investigação em Matemática e Aplicações (CIMA-UÉ)ÉvoraPortugal
  2. 2.Departamento de Matemática and CAMGSD - Centro de Análise Matemática, Geometria e Sistemas Dinâmicos, Instituto Superior TécnicoUniversity of LisbonLisboaPortugal

Personalised recommendations