The local structure of Poisson manifolds

Journal of Differential Geometry

Volumn 18, Issue 3, 1983, Pages 523-557

  Weinstein, Alan ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84972531238     PISSN: 0022040X     EISSN: 1945743X     Source Type: Journal    
DOI: 10.4310/jdg/1214437787     Document Type: Article

References (48)

1. R. Abraham & J. Marsden, Foundations of mechanics, 2nd edition, Benjamin Cummings, Reading, Massaschusetts, 1978.
2. R. Abraham & J. Robbin, Transversal mappings and flows, Benjamin, New York, 1967.
3. V. I. Arnold, Mathematical methods of classical mechanics, Graduate Text in Math., Vol. 60, Springer, New York, 1978.
4. F. A. Berezin, Some remarks about the associated envelope of a Lie algebra, Funct. Anal. Appl. 1(1967)91-102.
5. F. A. Berezin, Quantization, Izv. Akad. Nauk. SSSR, Ser. Matem. 38 (1974) 1116-1175 (= Math. USSR Izv. 38 (1974) 1109-1164).
6. R. Berger, Géométrie algébrique de Poisson et deformations, Publ. Dept. Math. Univ. Claude Bernard, Lyon 16, fasc. 2, 1979, 1-69.
7. R. L. Bishop & R. J. Crittenden, Geometry of manifolds, Academic Press, New York, 1964.
8. M. Duflo & M. Vergne, Unepropriėté de la representation coadjointe d'une algebre de Lie, C. R. Acad. Sci. Paris, Sér. A-B, 268 (1969) A583-A585.
9. H. Freudenthal & H. deVries, Linear Lie groups, Academic Press, New York, 1969.
10. J. Gibbons, Collisionless Boltzmann equations and integrable moment equations, Physica 3D (1979) 503-511.
11. M. Golubitsky & V. Guillemin, Stable mappings and their singularities, Graduate Texts in Math., Vol. 14, Springer, New York, 1973.
12. M. Gotay, R. Lashof, J. Sniatycki & A. Weinstein, Closed forms on symplectic fibre bundles, Comment. Math. Helv., to appear.
13. V. W. Guillemin, Band asymptotics in two dimensions, Advances in Math. 42 (1981) 248-282.
14. V. Guillemin & S. Sternberg, Remarks on a paper of Hermann, Trans. Amer. Math. Soc. 130 (1968) 110-116.
15. V. Guillemin & S. Sternberg, The moment map and collective motion, Ann. Physics 127 (1980) 220-253.
16. S. Helgason, Differential geometry and symmetric spaces, Academic Press, New York, 1962.
17. R. Hermann, Cartan connections and the equivalence problem for geometric structures, Contributions to Differential Equations 3 (1964) 199-248.
18. R. Hermann, The formal linearization of a semisimple Lie algebra of vector fields about a singular point, Trans. Amer. Math. Soc. 130 (1968) 105-109.
19. R. Hermann, Gauge fields and Cartan-Ehresmann connections. Part A: Interdisciplinary mathe matics, Vol. X, Math. Sci. Press, Brookline, 1975.
20. R. Hermann, Toda lattices, cosymplectic manifolds, Bäcklund transformations and kinks. Part A: Interdisciplinary mathematics, Vol. XV, Math. Sci. Press, Brookline, 1977.
21. P. J. Holmes & J. E. Marsden, Horseshoes and Arnold diffusion for Hamiltonian systems on Lie groups, Indiana Univ. Math. 32 (1983) 273-309.
22. N. Jacobson, Lie algebras, Interscience, New York, 1962.
23. D. Kazhdan, B. Kostant & S. Sternberg, Hamiltonian group actions and dynamical systems of Calogero type, Comm. Pure Appl. Math. 31 (1978) 481-508.
24. A. A. Kirillov, Unitary representations of nilpotent Lie groups, Russian Math. Surveys 17 (1962) 53-104 (= Usp. Mat. Nauk. 17 (1962) 57-110).
25. A. A. Kirillov, Local Lie algebras, Russian Math. Surveys 31 (1976) 56-75 (= Uspekhi Mat. Nauk. 31(1976) 57-76).
26. B. Kostant, Orbits, symplectic structures, and representation theory, Proc. U.S.-Japan Seminar in Differential Geometry, Kyoto, Japan, 1965, Nippon Hyoronisha Tokyo, 1965, 71.
27. B. Kostant, Quantization and unitary representations. Part I: Prequantization, Lecture Notes in Math. Vol. 170, Springer, Berlin, 1970, 87-208.
28. A. Lichnerowicz, Les varieties de Poisson et leurs algèbres de Lie associèes, J. Differential Geometry 12 (1977) 253-300.
29. S. Lie, Theorie der transformationsgruppen, (Zweiter Abschnitt, unter mitwirkung von Prof. Dr. Friedrich Engel), Teubner, Leipzig, 1890.
30. B. Malgrange, Equation de Lie. II, J. Differential Geometry 7 (1972) 117-141.
31. J. Marsden & A. Weinstein, Reduction of symplectic manifolds with symmetry, Rep. Mathe matical Phys. 5 (1974) 121-130.
32. J. Marsden & A. Weinstein, Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids, Proc. of the Conference on Order in Chaos, Los Alamos, 1982, A. Scott, ed., Physica, to appear.
33. J. Marsden, T. Ratiu & A. Weinstein, Semi-direct products and reduction in mechanics, to appear in Trans. Amer. Math. Soc.
34. A. S. Mishchenko & A. T. Fomenko, Generalized Liouville method of integration of Hamilto nian systems, Funct. Anal. Appl. 12 (1978) 113-121 (= Funkt. Anal. Prilozh. 12 (1978) 46-56).
35. P. J. Morrison, The Maxwell-Vlasov equations as a continuous Hamiltonian system, Phys. Lett. 80A (1980) 383-386.
36. J. Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965) 286-294.
37. R. Moussu, Sur l' existence d 'integrates premieres pour un germe de forme de Pfaff, Ann. Inst. Fourier (Grenoble) 26 (1976) 171-120.
38. G. Reeb, Sur certaines propriètès topologiques des variétés feuiłletées, Hermann, Paris, 1952.
39. J. Roels & A. Weinstein, On functions whose Poisson brackets are constant, J. Mathematical Phys. 12(1971) 1482-1486.
40. W. Schmid, Poincaré and Lie groups, Bull. Amer. Math. Soc. 6 (1982) 175-186.
41. J. M. Souriau, Quantification géometrique, Comm. Math. Phys. 1 (1966) 374-398.
42. J. A. Schouten, Über differentialkomitanten zweier kontravarianter Grössen, Nederl. Akad. Wetensch. Proc. Ser. A, 43 (1940) 449-452.
43. M. Tabor & Y. M. Treve (eds.), Mathematical methods in hydro-dynamics and integrability in dynamical systems, Proc. Amer. Inst. Phys. Conference 88, 1982.
44. W. M. Tulczyjew, Poisson brackets and canonical manifolds, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. 22 (1974) 931-934.
45. J. Vey, Sur le lemme de Morse, Invent. Math. 40 (1977) 1-10.
46. A. M. Vinogradov & I. S. Krasilshchik, What is the Hamiltonian formalism? Russian Math. Surveys 30 (1975) 117-202 (= Uspehkhi Mat. Nauk. 30 (1975) 173-198).
47. A. Weinstein, Symplectic manifolds and their lagrangian submanifolds, Advances in Math. 6 (1971) 329-346.
48. A. Weinstein, Fat bundles and symplectic manifolds, Advances in Math. 37 (1971) 239-250.