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June  2013, 5(2): 233-256. doi: 10.3934/jgm.2013.5.233

Twisted isotropic realisations of twisted Poisson structures

Nicola Sansonetto 1, and  Daniele Sepe 2,

1. 

Dipartimento di Informatica, Università degli Studi di Verona, Ca' Vignal 2, Strada Le Grazie 15, 37134 Verona,, Italy
2. 

CAMGSD, Instituto Superior Técnico, Av. Rovisco Pais, Lisboa, 1049-001, Portugal

Received  July 2012 Revised  March 2013 Published  July 2013

Motivated by the recent connection between nonholonomic integrable systems and twisted Poisson manifolds made in [3], this paper investigates the global theory of integrable Hamiltonian systems on almost symplectic manifolds as an initial step to understand Hamiltonian integrability on twisted Poisson (and Dirac) manifolds. Non-commutative integrable Hamiltonian systems on almost symplectic manifolds were first defined in [17], which proved existence of local generalised action-angle coordinates in the spirit of the Liouville-Arnol'd theorem. In analogy with their symplectic counterpart, these systems can be described globally by twisted isotropic realisations of twisted Poisson manifolds, a special case of symplectic realisations of twisted Dirac structures considered in [8]. This paper classifies twisted isotropic realisations up to smooth isomorphism and provides a cohomological obstruction to the construction of these objects, generalising some of the main results of [13].
Keywords: almost symplectic manifolds., Twisted Poisson manifolds, isotropic realisations, Hamiltonian integrable systems.
Mathematics Subject Classification: 37J35, 53D15, 53D1.
Citation: Nicola Sansonetto, Daniele Sepe. Twisted isotropic realisations of twisted Poisson structures. Journal of Geometric Mechanics, 2013, 5 (2) : 233-256. doi: 10.3934/jgm.2013.5.233
References:
[1]

M. Adler, P. van Moerbeke and P. Vanhaecke, "Algebraic integrability, Painlevé Geometry and Lie Algebras,'' Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, A Series of Modern Surveys in Mathematics, 47, Springer-Verlag, Berlin, 2004.  Google Scholar

[2]

P. Ashwin and I. Melbourne, Noncompact drift for relative equilibria and relative periodic orbits, Nonlinearity, 10 (1997), 595-616. doi: 10.1088/0951-7715/10/3/002.  Google Scholar

[3]

P. Balseiro and L. García-Naranjo, Gauge transformations, twisted Poisson brackets and Hamiltonianization of nonholonomic systems, Arch. Rat. Mech. Anal., 205 (2012), 267-310. doi: 10.1007/s00205-012-0512-9.  Google Scholar

[4]

L. Bates and R. H. Cushman, What is a completely integrable nonholonomic dynamical system?, in "Proceedings of the XXX Symposium on Mathematical Physics'' (Toruń, 1998), Rep. Math. Phys., 44 (1999), 29-35. doi: 10.1016/S0034-4877(99)80142-6.  Google Scholar

[5]

G. Blankenstein and T. S. Ratiu, Singular reduction of implicit Hamiltonian systems, Rep. Math. Phys., 53 (2004), 211-260. doi: 10.1016/S0034-4877(04)90013-4.  Google Scholar

[6]

A. M. Bloch and D. V. Zenkov, Dynamics of the n-Dimensional Suslov problem, J. Geom. Phys., 34 (2000), 121-136. doi: 10.1016/S0393-0440(99)00058-3.  Google Scholar

[7]

O. I. Bogoyavlenskij, Extended integrability and bi-Hamiltonian systems, Comm. Math. Phys., 196 (1998), 19-51. doi: 10.1007/s002200050412.  Google Scholar

[8]

H. Bursztyn, M. Crainic, A. Weinstein and C. Zhu, Integration of twisted Dirac brackets, Duke Math. J., 123 (2004), 549-607. doi: 10.1215/S0012-7094-04-12335-8.  Google Scholar

[9]

A. S. Cattaneo and P. Xu, Integration of twisted Poisson structures, J. Geom. Phys., 49 (2004), 187-196. doi: 10.1016/S0393-0440(03)00086-X.  Google Scholar

[10]

M. Crainic and R. L. Fernandes, Integrability of Lie brackets, Ann. of Math. (2), 157 (2003), 575-620. doi: 10.4007/annals.2003.157.575.  Google Scholar

[11]

M. Crainic and R. L. Fernandes, Integrability of Poisson brackets, J. Diff. Geom., 66 (2004), 71-137.  Google Scholar

[12]

R. H. Cushman and J. J. Duistermaat, Non-Hamiltonian monodromy, J. Diff. Eq., 172 (2001), 42-58. doi: 10.1006/jdeq.2000.3852.  Google Scholar

[13]

P. Dazord and P. Delzant, Le problème général des variables actions-angles, J. Diff. Geom., 26 (1987), 223-251.  Google Scholar

[14]

J. B. Delos, G. Dhont, D. A. Sadovskií and B. I. Zhilinskií, Dynamical manifestations of Hamiltonian monodromy, Ann. Physics, 324 (2009), 1953-1982. doi: 10.1016/j.aop.2009.03.008.  Google Scholar

[15]

J. J. Duistermaat, On global action-angle coordinates, Comm. Pure Appl. Math., 33 (1980), 687-706. doi: 10.1002/cpa.3160330602.  Google Scholar

[16]

A. El Kacimi-Alaoui, Sur la cohomologie feuilletée, Compositio Math., 49 (1983), 195-215.  Google Scholar

[17]

F. Fassò and N. Sansonetto, Integrable almost-symplectic Hamiltonian systems, J. Math. Phys., 48 (2007), 092902, 13 pp. doi: 10.1063/1.2783937.  Google Scholar

[18]

F. Fassò, Superintegrable Hamiltonian systems: Geometry and perturbations, Acta Appl. Math., 87 (2005), 93-121. doi: 10.1007/s10440-005-1139-8.  Google Scholar

[19]

, F. Fassò, A. Giacobbe, L. Garcia-Naranjo and N. Sansonetto,, in preparation., ().   Google Scholar

[20]

F. Fassò, F. and A. Giacobbe, Geometry of invariant tori of certain integrable systems with symmetry and an application to a nonholonomic system, SIGMA Symmetry Integrability Geom. Methods Appl., 3 (2007), 051, 12 pp. doi: 10.3842/SIGMA.2007.051.  Google Scholar

[21]

M. Field, Equivariant dynamical systems, Trans. Amer. Math. Soc., 259 (1980), 185-205. doi: 10.1090/S0002-9947-1980-0561832-4.  Google Scholar

[22]

W. M. Goldman, M. W. Hirsch and G. Levitt, Invariant measures for affine foliations, Proc. Amer. Math. Soc., 86 (1982), 511-518. doi: 10.1090/S0002-9939-1982-0671227-8.  Google Scholar

[23]

J. Hermans, A symmetric sphere rolling on a surface, Nonlinearity, 8 (1995), 493-515. doi: 10.1088/0951-7715/8/4/003.  Google Scholar

[24]

C. Klimčík and T. Strobl, WZW-Poisson manifolds, J. Geom. Phys., 43 (2002), 341-344. doi: 10.1016/S0393-0440(02)00027-X.  Google Scholar

[25]

Y. Kosmann-Schwarzbach, Quasi, twisted, and all that... in Poisson geometry and Lie algebroid theory, in "The Breadth of Symplectic and Poisson Geometry," Progress in Mathematics, 232, Birkhäuser Boston, Boston, MA, (2005), 363-389. doi: 10.1007/0-8176-4419-9_12.  Google Scholar

[26]

C. Laurent-Gengoux, E. Miranda and P. Vanhaecke, Action-angle coordinates for integrable systems on Poisson manifolds, Int. Math. Res. Not., 8 (2011), 1839-1869. doi: 10.1093/imrn/rnq130.  Google Scholar

[27]

K. C. H. Mackenzie, "General Theory Of Lie Groupoids and Lie Algebroids,'' London Mathematical Society Lecture Notes Series, 213, Cambridge University Press, Cambridge, 2005.  Google Scholar

[28]

J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems,'' Texts in Applied Mathematics, 17, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-2682-6.  Google Scholar

[29]

A. S. Miščenko and A. T, Fomenko, Integration of Hamiltonian systems with noncommutative symmetries, Trudy Sem. Vektor. Tenzor. Anal., 20 (1981), 5-54.  Google Scholar

[30]

I. Moerdijk and J. Mrčun, "Introduction to Foliations and Lie Groupoids,'' Cambridge Studies in Advanced Mathematics, 91, Cambridge University Press, Cambridge, 2003. doi: 10.1017/CBO9780511615450.  Google Scholar

[31]

N. N. Nehorošev, Action-angle variables and their generalizations, (Russian) Trudy Moskov. Mat. Obšč, 26 (1972), 180-198.  Google Scholar

[32]

J.-S. Park, Topological open p-branes, in "Symplectic Geometry and Mirror Symmetry'' (Seoul, 2000), World Sci. Publ., River Edge, NJ, (2001), 311-384. doi: 10.1142/9789812799821_0010.  Google Scholar

[33]

A. Pelayo, T. S. Ratiu and S. Vũ Ngoc, Symplectic bifurcation theory for integrable systems,, preprint, ().   Google Scholar

[34]

C. A. Rossi, Principal bundles with groupoid structure: Local vs. global theory and nonabelian Čech cohomology,, preprint, ().   Google Scholar

[35]

, N. Sansonetto and D. Sepe,, in preparation., ().   Google Scholar

[36]

D. Sepe, Almost Lagrangian obstruction, Diff. Geom. Appl., 29 (2011), 787-800. doi: 10.1016/j.difgeo.2011.08.007.  Google Scholar

[37]

P. Ševera and A. Weinstein, Poisson geometry with a 3-form background, in "Noncommutative Geometry and String Theory'' (Yokohama, 2001), Progr. Theoret. Phys. Suppl., 144 (2001), 145-154. doi: 10.1143/PTPS.144.145.  Google Scholar

[38]

I. Vaisman, "Lectures on the Geometry of Poisson Manifolds,'' Progress in Mathematics, 118, Birkäuser Verlag, Basel, 1994. doi: 10.1007/978-3-0348-8495-2.  Google Scholar

[39]

A. Weinstein, Symplectic groupoids and Poisson manifolds, Bull. Amer. Math. Soc., 16 (1988), 101-104. doi: 10.1090/S0273-0979-1987-15473-5.  Google Scholar

[40]

D. V. Zenkov, The geometry of the Routh problem, Jour. Nonlin. Science, 5 (1995), 503-519. doi: 10.1007/BF01209025.  Google Scholar

show all references

References:
[1]

M. Adler, P. van Moerbeke and P. Vanhaecke, "Algebraic integrability, Painlevé Geometry and Lie Algebras,'' Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, A Series of Modern Surveys in Mathematics, 47, Springer-Verlag, Berlin, 2004.  Google Scholar

[2]

P. Ashwin and I. Melbourne, Noncompact drift for relative equilibria and relative periodic orbits, Nonlinearity, 10 (1997), 595-616. doi: 10.1088/0951-7715/10/3/002.  Google Scholar

[3]

P. Balseiro and L. García-Naranjo, Gauge transformations, twisted Poisson brackets and Hamiltonianization of nonholonomic systems, Arch. Rat. Mech. Anal., 205 (2012), 267-310. doi: 10.1007/s00205-012-0512-9.  Google Scholar

[4]

L. Bates and R. H. Cushman, What is a completely integrable nonholonomic dynamical system?, in "Proceedings of the XXX Symposium on Mathematical Physics'' (Toruń, 1998), Rep. Math. Phys., 44 (1999), 29-35. doi: 10.1016/S0034-4877(99)80142-6.  Google Scholar

[5]

G. Blankenstein and T. S. Ratiu, Singular reduction of implicit Hamiltonian systems, Rep. Math. Phys., 53 (2004), 211-260. doi: 10.1016/S0034-4877(04)90013-4.  Google Scholar

[6]

A. M. Bloch and D. V. Zenkov, Dynamics of the n-Dimensional Suslov problem, J. Geom. Phys., 34 (2000), 121-136. doi: 10.1016/S0393-0440(99)00058-3.  Google Scholar

[7]

O. I. Bogoyavlenskij, Extended integrability and bi-Hamiltonian systems, Comm. Math. Phys., 196 (1998), 19-51. doi: 10.1007/s002200050412.  Google Scholar

[8]

H. Bursztyn, M. Crainic, A. Weinstein and C. Zhu, Integration of twisted Dirac brackets, Duke Math. J., 123 (2004), 549-607. doi: 10.1215/S0012-7094-04-12335-8.  Google Scholar

[9]

A. S. Cattaneo and P. Xu, Integration of twisted Poisson structures, J. Geom. Phys., 49 (2004), 187-196. doi: 10.1016/S0393-0440(03)00086-X.  Google Scholar

[10]

M. Crainic and R. L. Fernandes, Integrability of Lie brackets, Ann. of Math. (2), 157 (2003), 575-620. doi: 10.4007/annals.2003.157.575.  Google Scholar

[11]

M. Crainic and R. L. Fernandes, Integrability of Poisson brackets, J. Diff. Geom., 66 (2004), 71-137.  Google Scholar

[12]

R. H. Cushman and J. J. Duistermaat, Non-Hamiltonian monodromy, J. Diff. Eq., 172 (2001), 42-58. doi: 10.1006/jdeq.2000.3852.  Google Scholar

[13]

P. Dazord and P. Delzant, Le problème général des variables actions-angles, J. Diff. Geom., 26 (1987), 223-251.  Google Scholar

[14]

J. B. Delos, G. Dhont, D. A. Sadovskií and B. I. Zhilinskií, Dynamical manifestations of Hamiltonian monodromy, Ann. Physics, 324 (2009), 1953-1982. doi: 10.1016/j.aop.2009.03.008.  Google Scholar

[15]

J. J. Duistermaat, On global action-angle coordinates, Comm. Pure Appl. Math., 33 (1980), 687-706. doi: 10.1002/cpa.3160330602.  Google Scholar

[16]

A. El Kacimi-Alaoui, Sur la cohomologie feuilletée, Compositio Math., 49 (1983), 195-215.  Google Scholar

[17]

F. Fassò and N. Sansonetto, Integrable almost-symplectic Hamiltonian systems, J. Math. Phys., 48 (2007), 092902, 13 pp. doi: 10.1063/1.2783937.  Google Scholar

[18]

F. Fassò, Superintegrable Hamiltonian systems: Geometry and perturbations, Acta Appl. Math., 87 (2005), 93-121. doi: 10.1007/s10440-005-1139-8.  Google Scholar

[19]

, F. Fassò, A. Giacobbe, L. Garcia-Naranjo and N. Sansonetto,, in preparation., ().   Google Scholar

[20]

F. Fassò, F. and A. Giacobbe, Geometry of invariant tori of certain integrable systems with symmetry and an application to a nonholonomic system, SIGMA Symmetry Integrability Geom. Methods Appl., 3 (2007), 051, 12 pp. doi: 10.3842/SIGMA.2007.051.  Google Scholar

[21]

M. Field, Equivariant dynamical systems, Trans. Amer. Math. Soc., 259 (1980), 185-205. doi: 10.1090/S0002-9947-1980-0561832-4.  Google Scholar

[22]

W. M. Goldman, M. W. Hirsch and G. Levitt, Invariant measures for affine foliations, Proc. Amer. Math. Soc., 86 (1982), 511-518. doi: 10.1090/S0002-9939-1982-0671227-8.  Google Scholar

[23]

J. Hermans, A symmetric sphere rolling on a surface, Nonlinearity, 8 (1995), 493-515. doi: 10.1088/0951-7715/8/4/003.  Google Scholar

[24]

C. Klimčík and T. Strobl, WZW-Poisson manifolds, J. Geom. Phys., 43 (2002), 341-344. doi: 10.1016/S0393-0440(02)00027-X.  Google Scholar

[25]

Y. Kosmann-Schwarzbach, Quasi, twisted, and all that... in Poisson geometry and Lie algebroid theory, in "The Breadth of Symplectic and Poisson Geometry," Progress in Mathematics, 232, Birkhäuser Boston, Boston, MA, (2005), 363-389. doi: 10.1007/0-8176-4419-9_12.  Google Scholar

[26]

C. Laurent-Gengoux, E. Miranda and P. Vanhaecke, Action-angle coordinates for integrable systems on Poisson manifolds, Int. Math. Res. Not., 8 (2011), 1839-1869. doi: 10.1093/imrn/rnq130.  Google Scholar

[27]

K. C. H. Mackenzie, "General Theory Of Lie Groupoids and Lie Algebroids,'' London Mathematical Society Lecture Notes Series, 213, Cambridge University Press, Cambridge, 2005.  Google Scholar

[28]

J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems,'' Texts in Applied Mathematics, 17, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-2682-6.  Google Scholar

[29]

A. S. Miščenko and A. T, Fomenko, Integration of Hamiltonian systems with noncommutative symmetries, Trudy Sem. Vektor. Tenzor. Anal., 20 (1981), 5-54.  Google Scholar

[30]

I. Moerdijk and J. Mrčun, "Introduction to Foliations and Lie Groupoids,'' Cambridge Studies in Advanced Mathematics, 91, Cambridge University Press, Cambridge, 2003. doi: 10.1017/CBO9780511615450.  Google Scholar

[31]

N. N. Nehorošev, Action-angle variables and their generalizations, (Russian) Trudy Moskov. Mat. Obšč, 26 (1972), 180-198.  Google Scholar

[32]

J.-S. Park, Topological open p-branes, in "Symplectic Geometry and Mirror Symmetry'' (Seoul, 2000), World Sci. Publ., River Edge, NJ, (2001), 311-384. doi: 10.1142/9789812799821_0010.  Google Scholar

[33]

A. Pelayo, T. S. Ratiu and S. Vũ Ngoc, Symplectic bifurcation theory for integrable systems,, preprint, ().   Google Scholar

[34]

C. A. Rossi, Principal bundles with groupoid structure: Local vs. global theory and nonabelian Čech cohomology,, preprint, ().   Google Scholar

[35]

, N. Sansonetto and D. Sepe,, in preparation., ().   Google Scholar

[36]

D. Sepe, Almost Lagrangian obstruction, Diff. Geom. Appl., 29 (2011), 787-800. doi: 10.1016/j.difgeo.2011.08.007.  Google Scholar

[37]

P. Ševera and A. Weinstein, Poisson geometry with a 3-form background, in "Noncommutative Geometry and String Theory'' (Yokohama, 2001), Progr. Theoret. Phys. Suppl., 144 (2001), 145-154. doi: 10.1143/PTPS.144.145.  Google Scholar

[38]

I. Vaisman, "Lectures on the Geometry of Poisson Manifolds,'' Progress in Mathematics, 118, Birkäuser Verlag, Basel, 1994. doi: 10.1007/978-3-0348-8495-2.  Google Scholar

[39]

A. Weinstein, Symplectic groupoids and Poisson manifolds, Bull. Amer. Math. Soc., 16 (1988), 101-104. doi: 10.1090/S0273-0979-1987-15473-5.  Google Scholar

[40]

D. V. Zenkov, The geometry of the Routh problem, Jour. Nonlin. Science, 5 (1995), 503-519. doi: 10.1007/BF01209025.  Google Scholar

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