Integrable systems and group actions
Open Mathematics (2014)
- Volume: 12, Issue: 2, page 240-270
- ISSN: 2391-5455
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topEva Miranda. "Integrable systems and group actions." Open Mathematics 12.2 (2014): 240-270. <http://eudml.org/doc/269684>.
@article{EvaMiranda2014,
abstract = {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.},
author = {Eva Miranda},
journal = {Open Mathematics},
keywords = {Integrable system; Momentum map; Poisson manifold; Contact manifold; Symplectic manifold; Group action; integrable system; momentum map; contact manifold; symplectic manifold; group action},
language = {eng},
number = {2},
pages = {240-270},
title = {Integrable systems and group actions},
url = {http://eudml.org/doc/269684},
volume = {12},
year = {2014},
}
TY - JOUR
AU - Eva Miranda
TI - Integrable systems and group actions
JO - Open Mathematics
PY - 2014
VL - 12
IS - 2
SP - 240
EP - 270
AB - 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.
LA - eng
KW - Integrable system; Momentum map; Poisson manifold; Contact manifold; Symplectic manifold; Group action; integrable system; momentum map; contact manifold; symplectic manifold; group action
UR - http://eudml.org/doc/269684
ER -
References
top- [1] Albouy A., Projective dynamics and classical gravitation, Regul. Chaotic Dyn., 2008, 13(6), 525–542 http://dx.doi.org/10.1134/S156035470806004X Zbl1229.70031
- [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 Zbl0933.53035
- [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 Zbl1022.37038
- [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 Zbl1273.34043
- [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 Zbl0031.24803
- [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 Zbl1074.53066
- [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 Zbl1028.81026
- [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 Zbl0592.58025
- [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 Zbl1040.53035
- [15] Delzant T., Hamiltoniens périodiques et images convexes de l’application moment, Bull. Soc. Math. France, 1988, 116(3), 315–339 Zbl0676.58029
- [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 Zbl0808.58025
- [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 Zbl0439.58014
- [18] Eliasson L.H., Normal Forms for Hamiltonian Systems with Poisson Commuting Integrals, PhD thesis, University of Stockholm, 1984 Zbl0702.58024
- [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 Zbl0702.58024
- [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 Zbl1147.53068
- [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 Zbl0092.39301
- [23] Guillemin V., Ginzburg V., Karshon Y., Moment Maps, Cobordisms, and Hamiltonian Group Actions, Math. Surveys Monogr., 98, American Mathematical Society, Providence, 2004 Zbl1197.53002
- [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 Zbl1244.53093
- [25] Guillemin V., Miranda E., Pires A.R., Symplectic and Poisson geometry on b-manifolds, preprint available at http://arxiv.org/abs/1206.2020 Zbl1296.53159
- [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 Zbl1333.53115
- [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 Zbl0356.35080
- [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 Zbl0155.05701
- [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 Zbl0522.58021
- [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 Zbl1191.53058
- [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 Zbl0707.58026
- [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 Zbl0992.53062
- [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 Zbl1186.37067
- [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 Zbl1271.53074
- [38] Laurent-Gengoux C., Miranda E., Splitting theorem and integrable systems in Poisson manifolds (in preparation) Zbl1222.53087
- [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 Zbl1222.53087
- [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 Zbl1079.53118
- [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 Zbl0897.58016
- [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 Zbl0379.53011
- [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 Zbl0871.58045
- [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 Zbl62.1513.02
- [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 Zbl0017.04105
- [48] Miranda E., On Symplectic Linearization of Singular Lagrangian Foliations, PhD thesis, Universitat de Barcelona, 2003 Zbl1040.53035
- [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 Zbl1232.53071
- [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 Zbl1078.58007
- [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 Zbl1068.37041
- [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 Zbl1112.53062
- [58] Nest R., Tsygan B., Formal deformations of symplectic manifolds with boundary, J. Reine Angew. Math., 1996, 481, 27–54 Zbl0866.58038
- [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 Zbl0103.01802
- [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 Zbl1215.53071
- [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 Zbl1225.53074
- [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 Zbl1012.37041
- [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 Zbl1303.37018
- [67] Weinstein A., Lectures on symplectic manifolds, CBMS Regional Conf. Ser. in Math., 29, American Mathematical Society, Providence, 1977 Zbl0406.53031
- [68] Weinstein A., The local structure of Poisson manifolds, J. Differential Geom., 1983, 18(3), 523–557 Zbl0524.58011
- [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 Zbl63.1290.01
- [70] Woodhouse N.M.J., Geometric quantization, 2nd ed., Oxford Math. Monogr., Clarendon Press, Oxford University Press, New York, 1992 Zbl0747.58004
- [71] Zung N.T., Symplectic topology of integrable Hamiltonian systems I. Arnold-Liouville with singularities, Compositio Math., 1996, 101(2), 179–215 Zbl0936.37042
- [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
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