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INdAM Fellowship Programs in Mathematics and/or Applications cofunded by Marie Skłodowska-Curie Actions



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Giovanni Manno

INdAM-COFUND Outgoing fellow between 2012-07-01 and 2014-06-30

Project information

Title

GMVMADS: Natural Hamiltonian systems admitting polynomial first integrals

Outgoing host organisation                        Return host organisation

Friedrich-Schiller-Universitaet Jena                  Politecnico di Torino

07737 Jena, Germany                                           Corso Duca degli Abruzzi, 24

                                                                                   10129 Torino, Italy

http://www.uni-jena.de/                                     http://www.polito.it/

Abstract

The present project aims at solving some classical problems in Differential Geometry that are relevant also in Mathematical Physics.

In 1882 Sophus Lie formulated the problem of finding all 2-dim. metrics admitting a projective vector field, i.e. a vector
field whose local flow preserves geodesics without necessarily preserving the affine parameter. The set of the projective vector ields is a Lie algebra. Such problem has been tackled by the applicant, R. L. Bryant and V. Matveev and solved in the case when the projective action is locally transitive. However, a complete answer is still missing and it is an objective of the project.

The above problem lies into the more general problem of describing all “natural” Hamiltonian systems (i.e. the Hamiltonian is given by the sum of kinetic and potential energy) possessing some symmetries (we underline that, in the case when the
potential in equal to zero, the Hamiltonian system is the geodesic flow of some metric). An intriguing and hard problem would be to describe natural Hamiltonian systems admitting first integrals that are polynomial in momenta. Such problem is classical as its formulation dates back at least to Darboux. An objective of the project is to obtain a local description of the aforementioned systems at least in the case they admit a cubic integral.

Fellow information

Research interests

  • Metrics admitting geodesic symmetries
  • Contact geometry of Monge-Ampère equations
  • Symmetries, conservation laws and Baecklund transformations of PDEs
  • Finite and infinite-dimensional integrable systems

Previous positions, awards

  • September 2004-August 2008: Research Fellow at the Department of
    Mathematics “E. De Giorgi” , University of Salento
  • September 2008-June 2012: Research Fellow at the Department of
    Mathematics and Applications , University of Milano-Bicocca

Publications, preprints, other works

  1. S. Igonin, J. Van De Leur, G. Manno, V. Trushkov, Infinite-dimensional Prolongation Lie Algebras and
    Multicomponent Landau-Lifshitz Systems Associated with Higher Genus Curves
    , J. Geom. Phys., 68 no. 8 (2013) 1-€“26.
  2. G. Manno, G. Metafune, On the extendability of conformal vector fields of 2-dimensional manifolds, Differential Geom. Appl., 30 (2012), 365-369.
  3. D.V. Alekseevsky, R. Alonso Blanco, G. Manno, F. Pugliese, Monge-Ampère equations on (para-)Khäler manifolds:
    from characteristic subspaces to special Lagrangian submanifolds
    , Acta Appl. Math., 120, no. 2 (2012), 3–27.
  4. D.V. Alekseevsky, R. Alonso Blanco, G. Manno, F. Pugliese, Contact geometry of multidimensional Monge-Ampère
    equations: characteristics, intermediate integrals and solutions
    , Annales de l’Institut Fourier (Grenoble), 62,
    no. 2 (2012), 497-524.
  5. R. Alonso Blanco, G. Manno, F. Pugliese, Normal forms for lagrangian distributions on
    5-dimensional contact manifolds
    , Differential Geom. Appl., 27 (2009), no. 2,
    212–229.
  6. R. L. Bryant, G. Manno, V. S. Matveev, A solution of a problem of Sophus Lie: normal
    forms of two-dimensional metrics admitting two projective vector
    fields
    , Math. Ann., 340, no. 2 (2008), 437-463.
  7. G. Manno, On the geometry of Grassmannian equivalent connections, Adv. Geom., 8 (2008) 329-342.
  8. G. Manno, J. Pohjanpelto, R. Vitolo, Gauge invariance, charge conservation, and
    variational principles
    , J. Geom. Phys., 58 no. 8 (2008) 996-1006.
  9. G. Manno, R. Vitolo, Geometric aspects of higher order variational
    principles on submanifolds
    , Acta Appl. Math., 101 (2008) 215-229.
  10. R. Alonso Blanco, G. Manno, F. Pugliese, Contact relative differential invariants for
    non-generic parabolic Monge-Ampère equations
    , Acta Appl. Math., 101 (2008) 5-19.
  11. G. Manno, F. Oliveri, R. Vitolo, On differential equations characterized by their
    Lie point symmetries
    , J. Math. Anal. and Appl., 332, no. 2 (2007), 767-786.
  12. G. Manno, F. Oliveri, R. Vitolo, Differential equations uniquely determined by
    algebras of point symmetries
    , Theoret. and Math. Phys., 151, no. 3 (2007), 843-850.
  13. D. Catalano Ferraioli, G. Manno, F. Pugliese, Generalised symmetries of partial differential
    equations via complex transformations
    , Bull. Austr. Math. Soc., 76 (2007), 243-262.
  14. D. Catalano Ferraioli, G. Manno, F. Pugliese, Contact symmetries of the elliptic Euler-Darboux
    equation
    , Note Mat., no. 23, 2 (2004), 3–14.
  15. G. Manno, R. Vitolo, Relativistic mechanics, cosymplectic manifolds and symmetries, Note Mat., no. 23, 2 (2004), 157-171.
  16. G. Manno, F. Oliveri, R. Vitolo, Differential equations and Lie symmetries, Wascom 2007, 14th Conference on Waves and Stability
    in Continuous Media, World Sci. Publ., Hackensack, NJ, (2008),
    459-€“468.
  17. G. Manno, F. Oliveri, R. Vitolo, On the correspondence between differential
    equations and symmetry algebras
    , Symmetry and Perturbation Theory 2007, Featured
    contribution, World Scientific Publishing, Hackensack, NJ, (2008),
    164–171.
  18. R. Alonso Blanco, G. Manno, F. Pugliese, Contact geometry of parabolic Monge-Ampère
    equations
    , Symmetry and Perturbation Theory 2007, World
    Scientific Publishing, Hackensack, NJ, (2008).
  19. G. Manno, F. Oliveri, R. Vitolo, Lie remarkable PDEs, Asymptotic methods in nonlinear wave phenomena,
    World Scientific Publishing, Hackensack, NJ, (2007), 119-131.
  20. G. Manno, F. Oliveri, R. Vitolo, On an inverse problem in group analysis of PDEs, Lie-remarkable equations, Wascom 2005 13th Conference on Waves and Stability in Continuous Media, World Scientific Publishing, Hackensack, NJ, (2006), 420- 431.
  21. A. Kisselev, G. Manno, On the symmetry structure of the minimal surface
    equation
    , Proc. of the IX int. conf. on Diff. Geom. and its
    Appl., Prague (Czech Rep.), Matfyzpress (2005), 499-506.
  22. G. Manno, The geometry of the geodesic equation in the
    framework of jets of submanifolds
    , Conference proceedings of AIP, no. 729 (2004),
    207-217.
  23. G. Manno, R. Vitolo, Variational sequences on finite order jets of
    submanifolds
    , Proc. of the VIII int. conf. on Diff. Geom. and
    its Appl., Opava (Czech Rep.), Math. Publ., 3 (2001), 435-446.
  24. S. Igonin, G. Manno, Fundamental Lie algebras for multicomponent Landau-Lifshitz systems, Preprint, 31 pp., available at http://www.staff.science.uu.nl/~igoni101/preprints/fa_mll.pdf
  25. G. Manno, V.S. Matveev, Projectively homogeneous metrics near points where they are not
    projectively homogeneous
    , Preprint, 33 pp.
  26. G. De Matteis, G. Manno, Lie algebra symmetry analysis of the Helfrich and Willmore surface
    shape equations
    , Preprint, 24 pp.
  27. D.V. Alekseevsky, R. Alonso Blanco, G. Manno, F. Pugliese, Finding solutions of parabolic
    Monge-AmpÚre equations by using the geometry of Cartan fields
    , Preprint, 21 pp.
  28. A.J. Di Scala, G. Manno, On the extension of parallel sections of linear connections, Preprint, 8 pp.
  29. G. Manno, Classification of 2-dimensional metrics admitting
    one projective vector field
    , In preparation
  30. G. Manno, B. Kruglikov, Normal forms for generic
    C∞-smooth Legendrian distributions on 5-dim. contact
    manifolds
    , In preparation
  31. S. Igonin, G. Manno, Fundamental algebras for scalar evolution equations, In preparation
  32. D.V. Alekseevsky, R. Alonso Blanco, G. Manno, F. Pugliese, Homogeneous contact manifolds and invariant
    Monge-AmpÚre equations of Goursat type
    , In preparation
  33. G. Manno, F. Oliveri, G. Saccomandi, R. Vitolo, Ordinary differential equations characterized by
    their symmetries
    , In preparation

Conferences, schools, other events

  • Symmetries of Differential Equation
    Lecce (Italy), 2004
  • International Workshop on Global Analysis
    Ankara, 2004
  • Symmetries and Pertubation Theory
    Cala Gonone (Italy), 2004
  • Formal theory of PDEs and their applications
    Joensuu (Finland), 2006
  • Symmetries and Pertubation Theory
    Otranto (Italy), 2007
  • Geometry and Algebra of PDEs
    Tromsoe (Norway), 2007
  • Diff. Geometry and its Applications
    Olomouc (Czech Rep.), 2007
  • Geometry and Symmetries of Differential Equations
    Santa Marinella (Italy), 17-22 May 2010
  • Giornate di Geometria Algebrica e argomenti correlati X
    Gargnano del Garda (Italy), 25-29 May 2010
  • Cartan Connections, Geometry of Homogeneous Spaces, and Dynamics
    Vienna, 10-23 July 2011
  • The interaction of Geometry and Representation Theory. Exploring new frontiers
    Vienna, 3-14 Settembre 2012
  • Pseudogroups and Differential Equations
    Tromsoe (Norway), 13-16 March 2013
  • Differential Geometry and Its Applications conference
    Brno (Czech Rep.), 19-23 August, 2013
  • Workshop on Geometry of PDEs and Integrability
    Opava (Czech Rep.), 14-18 October, 2013


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