Skip to main content
SpringerLink
Log in

Some results on cosymplectic manifolds

  • Original Paper
  • Published:
Geometriae Dedicata Aims and scope Submit manuscript

Abstract

We obtain a generalization of the Kodaira-Morrow stability theorem for cosymplectic structures. We investigate cosymplectic geometry on Lie groups and on their compact quotients by uniform discrete subgroups. In this way we show that a compact solvmanifold admits a cosymplectic structure if and only if it is a finite quotient of a torus.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alvarez López J.A., Kordyukov Y.: Long time behavior of leafwise heat flow for Riemannian foliations. Compositio Math. 125, 129–153 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  2. Auslander L.: Discrete uniform subgroups of solvable Lie groups. Trans. Amer. Math. Soc. 99, 398–402 (1961)

    MathSciNet  MATH  Google Scholar 

  3. Baues O., Cortes D.: Aspherical Kähler manifolds with solvable fundamental group. Geom. Dedicata 122, 215–229 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Blair D.E.: Contact manifolds in Riemannian geometry. Lecture Notes in Math, vol. 509. Springer, Berlin (1976)

    Google Scholar 

  5. Blair D.E., Goldberg S.I.: Topology of almost contact manifolds. J. Differ. Geom. 1, 347–354 (1967)

    MathSciNet  MATH  Google Scholar 

  6. Chinea D., de León M., Marrero J.C.: Topology of cosymplectic manifolds. J. Math. Pures Appl. (9) 72(6), 567–591 (1993)

    MathSciNet  MATH  Google Scholar 

  7. Chinea D., de León M., Marrero J.C.: Spectral sequences on Sasakian and cosymplectic manifolds. Houston J. Math. 23, 631–649 (1997)

    MathSciNet  MATH  Google Scholar 

  8. Dacko P.: On almost cosymplectic manifolds with the structure vector field ξ belonging to the k-nullity distribution. Balkan J. Geom. Appl. 5(2), 47–60 (2000)

    MathSciNet  MATH  Google Scholar 

  9. Dacko P., Olszak Z.: On almost cosymplectic (−1, μ, 0)-spaces. Cent. Eur. J. Math. 3(2), 318–330 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. D’Atri J.E.: Holomorphic sectional curvatures of bounded homogeneous domains and related questions. Trans. Amer. Math. Soc. 256, 405–413 (1985)

    MathSciNet  Google Scholar 

  11. De León M., Marrero J.C.: Compact cosymplectic manifolds with transversally positive definite Ricci tensor. Rend. Mat. Appl. 17(7), 607–624 (1997)

    MathSciNet  MATH  Google Scholar 

  12. Deligne P., Griffiths P., Morgan J., Sullivan D.: Real homotopy theory of Kähler manifolds. Invent. Math. 29(3), 245–274 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  13. Deninger C., Singhof W.: Real polarizable Hodge structures arising from foliations. Ann. Global Anal. Geom. 21(4), 377–399 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  14. Dorfmeister J.: Homogeneous Kähler manifolds admitting a transitive solvable group of automorphisms. Annales scientifiques de l’École Normale Supérieure Sér. 4(18), 143–180 (1985)

    MathSciNet  Google Scholar 

  15. Dragomir S., Tomassini G.: Differential Geometry and Analysis on CR Manifolds, Progress in Mathematics, vol. 246. Birkhäuser Boston, Inc., Boston, MA (2006)

    Google Scholar 

  16. Fujitani T.: Complex-valued differential forms on normal contact Riemannian manifolds. Tohoku Math. J. 18(2), 349–361 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gindikin, S., Piateckii Sapiro, I.I., Vinberg, E.B.: Homogeneous Kähler manifolds, 1968, Geometry of homogeneous Bounded Domains, C.I.M.E., pp. 1–88 (1967)

  18. Hano J.: On Kählerian homogeneous manifolds of unimodular Lie groups. Amer. J. Math. 79, 885–900 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hasegawa K.: A note on compact solvmanifolds with Kähler structures. Osaka J. Math. 43, 131–135 (2006)

    MathSciNet  MATH  Google Scholar 

  20. Kim T.W., Pak H.K.: Canonical foliations of certain classes of almost contact metric structures. Acta Math. Sin. 21, 841–846 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kodaira, K., Morrow, J.: Complex manifolds. Reprint of the 1971 edition with errata. AMS Chelsea Publishing, Providence, RI (2006). x+194 pp.

  22. Lichnerowicz A., Medina A.: Groupes á structures symplectiques ou Kähleriennes invariantes. C. R. Acad. Sci. Paris Sér. I Math. 306(3), 133–138 (1988)

    MathSciNet  MATH  Google Scholar 

  23. Marrero J.C., Padron E.: New examples of compact cosymplectic solvmanifolds. Arch. Math. (Brno) 34(3), 337–345 (1998)

    MathSciNet  MATH  Google Scholar 

  24. Milnor J.: Curvatures of left invariant metrics on Lie groups. Adv. Math. 21(3), 293–329 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  25. Molino P.: Géométrie globale des feuilletages Riemanniens. Proc. Nederl. Acad. A 1(85), 45–76 (1982)

    MathSciNet  Google Scholar 

  26. Molino P.: Riemannian Foliations, Progr. in Math. 73. Birkhäuser, Basel (1988)

    Google Scholar 

  27. Pyateeskii-Shapiro I.I.: Automorphic Functions and the Geometry of Classical Domains, Mathematics and its Applications, vol. 8. Gordon and Breach, New York-London-Paris (1969)

    Google Scholar 

  28. Reinhart B.L.: Foliated manifolds with bundle-like metrics. Ann. Math. 69, 119–132 (1959)

    Article  MathSciNet  Google Scholar 

  29. Sullivan, D.: Differential Forms and the Topology of Manifolds. Manifolds, Tokyo 1973 (Proc. Internat. Conf. Tokyo, 1973), pp. 37–49. University Tokyo Press, Tokyo (1975)

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università di Torino, Via Carlo Alberto 10, 10123, Torino, Italy

    Anna Fino & Luigi Vezzoni

Authors
  1. Anna Fino
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Luigi Vezzoni
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Luigi Vezzoni.

Additional information

This work was supported by the Project M.I.U.R. “Riemann Metrics and Differenziable Manifolds” and by G.N.S.A.G.A. of I.N.d.A.M.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fino, A., Vezzoni, L. Some results on cosymplectic manifolds. Geom Dedicata 151, 41–58 (2011). https://doi.org/10.1007/s10711-010-9518-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10711-010-9518-3

Keywords

Mathematics Subject Classification (2000)

Search

Navigation