Integrable systems and group actions

• Autorzy: Eva Miranda
• Języki publikacji: EN

Abstrakty

EN

The main purpose of this paper is to present in a unified approach to different results concerning group actions and integrable systems in symplectic, Poisson and contact manifolds. Rigidity problems for integrable systems in these manifolds will be explored from this perspective.

Słowa kluczowe

EN

Integrable system; Momentum map; Poisson manifold; Contact manifold; Symplectic manifold; Group action

Kategorie tematyczne

• 58E40: Group actions
• 53D50: Geometric quantization
• 53D17: Poisson manifolds; Poisson groupoids and algebroids
• 37J35: Completely integrable systems, topological structure of phase space, integration methods

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2014
• Tom: 12
• Numer: 2
• Strony: 240-270

Daty

wydano

2014-02-01

online

2013-11-21

Twórcy

autor

Eva Miranda

• Universitat Politècnica de Catalunya

Bibliografia

• [1] Albouy A., Projective dynamics and classical gravitation, Regul. Chaotic Dyn., 2008, 13(6), 525–542 http://dx.doi.org/10.1134/S156035470806004X
• [2] Arnol’d V.I., Mathematical Methods of Classical Mechanics, Grad. Texts in Math., 60, Springer, New York-Heidelberg, 1978 http://dx.doi.org/10.1007/978-1-4757-1693-1
• [3] Banyaga A., The geometry surrounding the Arnold-Liouville theorem, In: Advances in Geometry, Progr. Math., 172, Birkhäuser, Boston, 1999
• [4] Banyaga A., Molino P., Géométrie des formes de contact complètement intégrables de type toriques, In: Séminaire Gaston Darboux de Géométrie et Topologie Différentielle, 1991–1992, Montpellier, Université Montpellier II, Montpellier, 1993, 1–25
• [5] Bolsinov A.V., Jovanovic B., Noncommutative integrability, moment map and geodesic flows, Ann. Global Anal. Geom., 2003, 23(4), 305–322 http://dx.doi.org/10.1023/A:1023023300665
• [6] Bolsinov A.V., Matveev V.S., Singularities of momentum maps of integrable Hamiltonian systems with two degrees of freedom, J. Math. Sci. (New York), 1999, 94(4), 1477–1500 http://dx.doi.org/10.1007/BF02365198
• [7] Chaperon M., Quelques outils de la théorie des actions différentiables, In: Third Schnepfenried Geometry Conference, 1, Schnepfenried, May 10–15, 1982, Astérisque, 107–108, Soc. Math. France, Paris, 1983, 259–275
• [8] Chaperon M., Normalisation of the smooth focus-focus: a simple proof, Acta Math. Vietnam., 2013, 38(1), 3–9 http://dx.doi.org/10.1007/s40306-012-0003-y
• [9] Chevalley C., Eilenberg S., Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc., 1948, 63(1), 85–124 http://dx.doi.org/10.1090/S0002-9947-1948-0024908-8
• [10] Colin de Verdière Y., Singular Lagrangian manifolds and semiclassical analysis, Duke Math. J., 2003, 116(2), 263–298 http://dx.doi.org/10.1215/S0012-7094-03-11623-3
• [11] Colin de Verdière Y., Vey J., Le lemme de Morse isochore, Topology, 1979, 18(4), 283–293 http://dx.doi.org/10.1016/0040-9383(79)90019-3
• [12] Colin de Verdière Y., Vũ Ngoc S., Singular Bohr-Sommerfeld rules for 2D integrable systems, Ann. Sci. Ècole Norm. Sup., 2003, 36(1), 1–55
• [13] Conn J.F., Normal forms for smooth Poisson structures, Ann. of Math., 1985, 121(3), 565–593 http://dx.doi.org/10.2307/1971210
• [14] Currás-Bosch C., Miranda E., Symplectic linearization of singular Lagrangian foliations in M 4, Differential Geom. Appl., 2003, 18(2), 195–205 http://dx.doi.org/10.1016/S0926-2245(02)00147-X
• [15] Delzant T., Hamiltoniens périodiques et images convexes de l’application moment, Bull. Soc. Math. France, 1988, 116(3), 315–339
• [16] Dufour J.-P., Molino P., Toulet A., Classification des systèmes intégrables en dimension 2 et invariants des modèles de Fomenko, C. R. Acad. Sci. Paris Sér. I Math., 1994, 318(10), 949–952
• [17] Duistermaat J.J., On global action-angle coordinates, Comm. Pure Appl. Math., 1980, 33(6), 687–706 http://dx.doi.org/10.1002/cpa.3160330602
• [18] Eliasson L.H., Normal Forms for Hamiltonian Systems with Poisson Commuting Integrals, PhD thesis, University of Stockholm, 1984
• [19] Eliasson L.H., Normal forms for Hamiltonian systems with Poisson commuting integrals - elliptic case, Comment. Math. Helv., 1990, 65, 4–35 http://dx.doi.org/10.1007/BF02566590
• [20] Fomenko A.T., Topological classification of all integrable Hamiltonian differential equations of general type with two degrees of freedom, In: The Geometry of Hamiltonian Systems, Berkeley, June 5–16, 1989, Math. Sci. Res. Inst. Publ., 22, Springer, New York, 1991, 131–339 http://dx.doi.org/10.1007/978-1-4613-9725-0_10
• [21] Geiges G., Contact geometry, In: Handbook of Differential Geometry, II, Elsevier/North-Holland, Amsterdam, 2006, 315–382 http://dx.doi.org/10.1016/S1874-5741(06)80008-7
• [22] Gray J.W., Some global properties of contact structures, Ann. of Math., 1959, 69(2), 421–450 http://dx.doi.org/10.2307/1970192
• [23] Guillemin V., Ginzburg V., Karshon Y., Moment Maps, Cobordisms, and Hamiltonian Group Actions, Math. Surveys Monogr., 98, American Mathematical Society, Providence, 2004
• [24] Guillemin V., Miranda E., Pires A.R., Codimension one symplectic foliations and regular Poisson structures, Bull. Braz. Math. Soc. (N.S.), 2011, 42(4), 607–623 http://dx.doi.org/10.1007/s00574-011-0031-6
• [25] Guillemin V., Miranda E., Pires A.R., Symplectic and Poisson geometry on b-manifolds, preprint available at http://arxiv.org/abs/1206.2020
• [26] Guillemin V., Miranda E., Pires A.R., Scott G., Toric actions on b-symplectic manifolds, preprint available at http://arxiv.org/abs/1309.1897
• [27] Guillemin V., Schaeffer D., On a certain class of Fuchsian partial differential equations, Duke Math. J., 1977, 44(1), 157–199 http://dx.doi.org/10.1215/S0012-7094-77-04408-8
• [28] Guillemin V.W., Sternberg S., Remarks on a paper of Hermann, Trans. Amer. Math. Soc., 1968, 130(1), 110–116 http://dx.doi.org/10.1090/S0002-9947-1968-0217226-9
• [29] Guillemin V., Sternberg S., The Gel’fand-Cetlin system and quantization of the complex flag manifolds, J. Funct. Anal., 1983, 52(1), 106–128 http://dx.doi.org/10.1016/0022-1236(83)90092-7
• [30] Hamilton M.D., Miranda E., Geometric quantization of integrable systems with hyperbolic singularities, Ann. Inst. Fourier (Grenoble), 2010, 60(1), 51–85 http://dx.doi.org/10.5802/aif.2517
• [31] Ito H., Action-angle coordinates at singularities for analytic integrable systems, Math. Z., 1991, 206(3), 363–407 http://dx.doi.org/10.1007/BF02571351
• [32] Karshon Y., Tolman S., Centered complexity one Hamiltonian torus actions, Trans. Amer. Math. Soc., 2001, 353(12), 4831–4861 http://dx.doi.org/10.1090/S0002-9947-01-02799-4
• [33] Khesin B., Tabaschnikov S., Contact complete integrability, Regul. Chaotic Dyn., 2010, 15(4–5), 504–520 http://dx.doi.org/10.1134/S1560354710040076
• [34] Knörrer H., Geodesics on quadrics and a mechanical problem of C. Neumann, J. Reine Angew. Math., 1982, 334, 69–78
• [35] Kostant B., On the definition of quantization, In: Géométrie Symplectique et Physique Mathématique, Colloq. Internat. CNRS, 237, Éditions Centre Nat. Recherche Sci., Paris, 1975, 187–210
• [36] Kruglikov B.S., Matveev V.S., Vanishing of the entropy pseudonorm for certain integrable systems, Electron. Res. Announc. Amer. Math. Soc., 2006, 12, 19–28 http://dx.doi.org/10.1090/S1079-6762-06-00156-9
• [37] Laurent-Gengoux C., Miranda E., Coupling symmetries with Poisson structures, Acta Math. Vietnam., 2013, 38(1), 21–32 http://dx.doi.org/10.1007/s40306-013-0008-1
• [38] Laurent-Gengoux C., Miranda E., Splitting theorem and integrable systems in Poisson manifolds (in preparation)
• [39] Laurent-Gengoux C., Miranda E., Vanhaecke P., Action-angle coordinates for integrable systems on Poisson manifolds, Int. Math. Res. Not. IMRN, 2011, 8, 1839–1869
• [40] Lerman E., Contact toric manifolds, J. Symplectic Geom., 2003, 1(4), 785–828 http://dx.doi.org/10.4310/JSG.2001.v1.n4.a6
• [41] Lerman E., Tolman S., Hamiltonian torus actions on symplectic orbifolds and toric varieties, Trans. Amer. Math. Soc., 1997, 349(10), 4201–4230 http://dx.doi.org/10.1090/S0002-9947-97-01821-7
• [42] Libermann P., Legendre foliations on contact manifolds, Differential Geom. Appl., 1991, 1(1), 57–76 http://dx.doi.org/10.1016/0926-2245(91)90022-2
• [43] Liouville J., Note sur l’intégration des équations différentielles de la dynamique, J. Math. Pures Appl., 1855, 20, 137–138
• [44] Lutz R., Sur la géométrie des structures de contact invariantes, Ann. Inst. Fourier (Grenoble), 1979, 29(1), 283–306 http://dx.doi.org/10.5802/aif.739
• [45] Matveev V.S., Integrable Hamiltonian systems with two degrees of freedom. Topological structure of saturated neighborhoods of points of focus-focus and saddle-saddle types, Sb. Math., 1996, 187(4), 495–524 http://dx.doi.org/10.1070/SM1996v187n04ABEH000122
• [46] Mineur H., Réduction des systèmes mécaniques à n degrés de liberté admettant n intégrales premières uniformes en involution aux systèmes à variable séparées, J. Math. Pures Appl., 1936, 15, 385–389
• [47] Mineur H., Sur les systèmes mécaniques dans lesquels figurent des paramètres fonctions du temps. Étude des systèmes admettant n intégrales premieres uniformes en involution. Extension à ces systèmes des conditions de quantification de Bohr-Sommerfeld, Le Journal de l’École Polytechnique, 1937, 143, 237–270
• [48] Miranda E., On Symplectic Linearization of Singular Lagrangian Foliations, PhD thesis, Universitat de Barcelona, 2003
• [49] Miranda E., A normal form theorem for integrable systems on contact manifolds, In: Proceedings of XIII Fall Workshop on Geometry and Physics, Murcia, September 20–22, 2004, Publ. R. Soc. Mat. Esp., 9, Real Sociedad Matemàtica Española, Madrid, 2005, 240–246
• [50] Miranda E., Some rigidity results for symplectic and Poisson group actions, In: XV International Workshop on Geometry and Physics, Puerto de la Cruz, September 11–16, 2006, Publ. R. Soc. Mat. Esp., 11, Real Sociedad Matemàtica Española, Madrid, 2007, 177–183
• [51] Miranda E., From action-angle coordinates to geometric quantization: a 30 minute round-trip, In: Geometric Quantization in the Non-Compact Setting, Oberwolfach, February 13–19, 2011, Oberwolfach Rep., 2011, 8(1), 425–521
• [52] Miranda E., Symplectic linearization of semisimple Lie algebra actions, manuscript
• [53] Miranda E., Symplectic equivalence of non-degenerate integrable systems (in preparation)
• [54] Miranda E., Monnier P., Zung N.T., Rigidity of Hamiltonian actions on Poisson manifolds, Adv. Math., 2012, 229(2), 1136–1179 http://dx.doi.org/10.1016/j.aim.2011.09.013
• [55] Miranda E., Vũ Ngoc S., A singular Poincaré lemma, Int. Math. Res. Not., 2005, 1, 27–45 http://dx.doi.org/10.1155/IMRN.2005.27
• [56] Miranda E., Zung N.T., Equivariant normal form for nondegenerate singular orbits of integrable Hamiltonian systems, Ann. Sci. École Norm. Sup., 2004, 37(6), 819–839
• [57] Miranda E., Zung N.T., A note on equivariant normal forms of Poisson structures, Math. Res. Lett., 2006, 13(5–6), 1001–1012 http://dx.doi.org/10.4310/MRL.2006.v13.n6.a14
• [58] Nest R., Tsygan B., Formal deformations of symplectic manifolds with boundary, J. Reine Angew. Math., 1996, 481, 27–54
• [59] Palais R.S., On the existence of slices for actions of non-compact Lie groups, Ann. of Math., 1961, 73, 295–323 http://dx.doi.org/10.2307/1970335
• [60] Paternain G.P., On the topology of manifolds with completely integrable geodesic flows. II, J. Geom. Phys., 1994, 13(2), 289–298 http://dx.doi.org/10.1016/0393-0440(94)90036-1
• [61] Pelayo A., Vũ Ngoc S., Semitoric integrable systems on symplectic 4-manifolds, Invent. Math., 2009, 177(3), 571–597 http://dx.doi.org/10.1007/s00222-009-0190-x
• [62] Pelayo Á., Vũ Ngoc S., Constructing integrable systems of semitoric type, Acta Math., 2011, 206(1), 93–125 http://dx.doi.org/10.1007/s11511-011-0060-4
• [63] Sniatycki J., On cohomology groups appearing in geometric quantization, In: Differential Geometric Methods in Mathematical Physics, Bonn, July 1–4, 1975, Lecture Notes in Math., 570, Springer, Berlin, 1977, 46–66 http://dx.doi.org/10.1007/BFb0087781
• [64] Vey J., Sur le lemme de Morse, Invent. Math., 1977, 40(1), 1–9 http://dx.doi.org/10.1007/BF01389858
• [65] Vũ Ngoc S., On semi-global invariants for focus-focus singularities, Topology, 2003, 42(2), 365–380 http://dx.doi.org/10.1016/S0040-9383(01)00026-X
• [66] Vũ Ngoc S., Wacheux C., Smooth normal forms for integrable Hamiltonian systems near a focus-focus singularity, Acta Math. Vietnam., 2013, 38(1), 107–122 http://dx.doi.org/10.1007/s40306-013-0012-5
• [67] Weinstein A., Lectures on symplectic manifolds, CBMS Regional Conf. Ser. in Math., 29, American Mathematical Society, Providence, 1977
• [68] Weinstein A., The local structure of Poisson manifolds, J. Differential Geom., 1983, 18(3), 523–557
• [69] Williamson J., On the algebraic problem concerning the normal forms of linear dynamical systems, Amer. J. Math., 1936, 58(1), 141–163 http://dx.doi.org/10.2307/2371062
• [70] Woodhouse N.M.J., Geometric quantization, 2nd ed., Oxford Math. Monogr., Clarendon Press, Oxford University Press, New York, 1992
• [71] Zung N.T., Symplectic topology of integrable Hamiltonian systems I. Arnold-Liouville with singularities, Compositio Math., 1996, 101(2), 179–215
• [72] Zung N.T., Symplectic topology of integrable Hamiltonian systems II. Topological classification, Compositio Math., 2003, 138(2), 125–156 http://dx.doi.org/10.1023/A:1026133814607

Identyfikatory

DOI

10.2478/s11533-013-0333-6