Skip to main content
SpringerLink
Log in

On Deformation of Poisson Manifolds of Hydrodynamic Type

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We study a class of deformations of infinite-dimensional Poisson manifolds of hydrodynamic type which are of interest in the theory of Frobenius manifolds. We prove two results. First, we show that the second cohomology group of these manifolds, in the Poisson-Lichnerowicz cohomology, is ‘‘essentially’’ trivial. Then, we prove a conjecture of B. Dubrovin about the triviality of homogeneous formal deformations of the above manifolds.

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. Dubrovin, B., Krichever, I.M., Novikov, S.P.: Integrable systems I. In: Encyclopaedia of Mathematical Sciences, 4, Dynamical systems IV, Berlin-Heidelberg-New York: Springer-Verlag, 1990, pp. 173–280

  2. Personal communication during the ‘‘Intensive INDAM two-month seminar on the Geometry of Frobenius manifolds’’, Milano, 2000. See also; B. Dubrovin, Y. Zhang,: Bi-Hamiltonian hierarchies in 2D topological field theory at one-loop approximation. Commun. Math. Phys. 198, 311–361 (1998)

    Article  MathSciNet  Google Scholar 

  3. Lichnerowicz, A.: Les varietes de Poisson et leurs algebres de Lie associees. J. Diff. Geom. 12, 253–300 (1977)

    MATH  Google Scholar 

  4. Magri, F., Morosi, C.: A Geometrical Characterization of integrable Hamiltonian Systems through the Theory of Poisson-Nijenhuis Manifolds. Quaderno S 19/1984 of the Department of Mathematics of the University of Milano

  5. Tonti, E.: Inverse problem: its general solution. Differential geometry, calculus of variations, and their applications. In: Lecture Notes in Pure and Appl. Math. 100, Berlin-Heidelberg-New York: Springer, 1985, pp. 497–510

  6. Vainberg, M.M.: Vatiational Methods for the Study of Nonlinear Operators. Holden-Day, 1964

  7. Vaisman, I.: Lectures on the geometry of Poisson manifolds. Basel: Birkhäuser Verlag, 1994

  8. Volterra, V.: Fonctions de lignes. Paris: Gauthier-Villars, 1913

  9. Volterra, V.: Theory of functionals and of integral and integro-differential equations. New York: Dover Publications Inc., 1959

Download references

Author information

Authors and Affiliations

  1. Dottorato in Matematica, University of Torino, via C. Alberto 10, 10123, Torino, Italy

    Luca Degiovanni

  2. Department of Mathematics and Applications, University of Milano Bicocca, via degli Arcimboldi 8, 20126, Milano, Italy

    Franco Magri

  3. Dottorato in Matematica, University of Palermo, via Archirafi 34, 90123, Palermo, Italy

    Vincenzo Sciacca

Authors
  1. Luca Degiovanni
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Franco Magri
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Vincenzo Sciacca
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Luca Degiovanni.

Additional information

Communicated by N.A. Nekrasov

Work sponsored by the Italian Ministry of Research under the project 40%: Geometry of Integrable Systems.

Acknowledgements. We sincerely thank B. Dubrovin for introducing us to the problem of deformation of Poisson manifolds of hydrodynamic type. We also thank G. Falqui for many useful discussions. We finally thank the Istituto Nazionale di Alta Matematica of Rome, who supported a meeting on the geometry of Frobenius manifolds, giving us the occasion to meet all together and discuss the problem.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Degiovanni, L., Magri, F. & Sciacca, V. On Deformation of Poisson Manifolds of Hydrodynamic Type. Commun. Math. Phys. 253, 1–24 (2005). https://doi.org/10.1007/s00220-004-1190-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-004-1190-8

Keywords

Search

Navigation