Noncommutative Integrability, Moment Map and Geodesic Flows

Annals of Global Analysis and Geometry

Volumn 23, Issue 4, 2003, Pages 305-322

  Bolsinov, Alexey V ^a   Jovanović, Božidar ^b  

^a MOSCOW STATE UNIVERSITY   (Russian Federation)

^b MATHEMATICAL INSTITUTE   (Serbia)

Author keywords

Geodesic flows; Hamiltonian action of a Lie group; Integrable Hamiltonian systems; Noncommutative integrability

Indexed keywords

EID: 0037259480     PISSN: 0232704X     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1023023300665     Document Type: Article

References (48)

1. Arnold, V. I.: Mathematical Methods of Classical Mechanics, Springer, New York, 1978.
2. Arnold, V. I., Kozlov, V. V. and Neishtadt, A. I.: Mathematical Aspects of Classical and Celestial Mechanics, Dynamical Systems III, Springer, New York, 1988.
3. Bazaikin, Ya. V.: Double quotients of Lie groups with an integrable geodesic flows, Sibirsk. Mat. Zh. 41(3) (2000), 513-530 [in Russian]. English translation: Siberian Math. J. 41(3) (2000), 419-432.
4. Bazaikin, Ya. V.: Double quotients of Lie groups with an integrable geodesic flows, Sibirsk. Mat. Zh. 41(3) (2000), 513-530 [in Russian]. English translation: Siberian Math. J. 41(3) (2000), 419-432.
5. Besse, A. L.: Manifolds All of Whose Geodesics Are Closed, Springer, New York, 1978.
6. Bolsinov, A. V.: Compatible Poisson brackets on Lie algebras and the completeness of families of functions in involution, Izv. Akad. Nauk SSSR, Ser. Mat. 55(1) (1991), 68-92 [in Russian]. English translation: Math. USSR-Izv. 38(1) (1992), 69-90.
7. Bolsinov, A. V.: Compatible Poisson brackets on Lie algebras and the completeness of families of functions in involution, Izv. Akad. Nauk SSSR, Ser. Mat. 55(1) (1991), 68-92 [in Russian]. English translation: Math. USSR-Izv. 38(1) (1992), 69-90.
8. Bolsinov, A. V. and Taimanov, I. A.: Integrable geodesic flow with positive topological entropy, Invent. Math. 140 (2000), 639-650.
9. Bolsinov, A. V. and Taimanov, I. A.: Integrable geodesic flows on the suspensions of toric automorphisms, Proc. Steklov. Inst. Math. 231 (2000), 46-63 [in Russian].
10. Bolsinov, A. V. and Jovanović, B.: Integrable geodesic flows on homogeneous spaces, Mat. Sb., 192(7) (2001), 21-40 [in Russian]. English translation: Sb. Mat. 192 (2001), 951-968
11. Bolsinov, A. V. and Jovanović, B.: Integrable geodesic flows on homogeneous spaces, Mat. Sb., 192(7) (2001), 21-40 [in Russian]. English translation: Sb. Mat. 192 (2001), 951-968
12. Brailov, A. V.: Complete integrability of some geodesic flows and integrable systems with non-commuting integrals, Dokl. Akad. Nauk SSSR 271(2) (1983), 273-276 [in Russian]. English translation: Soviet Math. Dokl. 28(1) (1983), 61-65.
13. Brailov, A. V.: Complete integrability of some geodesic flows and integrable systems with non-commuting integrals, Dokl. Akad. Nauk SSSR 271(2) (1983), 273-276 [in Russian]. English translation: Soviet Math. Dokl. 28(1) (1983), 61-65.
14. Brailov, A. V.: Construction of complete integrable geodesic flows on compact symmetric spaces, Izv. Akad. Nauk SSSR, Ser. Mat. 50(2) (1986), 661-674 [in Russian]. English translation: Math. USSR-Izv. 50(4) (1986), 19-31.
15. Brailov, A. V.: Construction of complete integrable geodesic flows on compact symmetric spaces, Izv. Akad. Nauk SSSR, Ser. Mat. 50(2) (1986), 661-674 [in Russian]. English translation: Math. USSR-Izv. 50(4) (1986), 19-31.
16. Butler, L.: A new class of homogeneous manifolds with Liouville-integrable geodesic flows, C. R. Math. Acad. Sci. Soc. Roy. Canada 21(4) (1999), 127-131.
17. Dazord, P. and Delzant, T.: Le problème général des variables actions-angles, J. Differential Geom. 26 (1987), 223-251.
18. Duran, C. E.: On the complete integrability of the geodesic flow of manifolds all of whose geodesics are closed, Ergodic Theory Dynamical Systems 17(6) (1997), 1359-1370.
19. Eschenburg, J. H.: Inhomogeneous spaces of positive curvature, Differential Geom. Appl. 2(2) (1992), 123-132.
20. Gromoll, D., Meyer, W.: An exotic sphere with nonnegative sectional curvature, Ann. of Math. 100 (1974), 401-406.
21. Guillemin, V. and Sternberg, S.: On collective complete integrability according to the method of Thimm, Ergodic Theory Dynamical Systems 3 (1983), 219-230.
22. Guillemin, V. and Sternberg, S.: Multiplicity-free spaces, J. Differential Geom. 19 (1984), 31-56.
23. Guillemin, V. and Sternberg, S.: Symplectic Techniques in Physics, Cambrige Univ. Press, Cambridge, 1984.
24. Kramer, M.: Sphärische Untergruppen in kompakten zusammenhangenden Liegruppen, Compositio Math. 38 (1979), 129-153.
25. Libermann, P. and Marle, C. M.: Symplectic Geometry and Analytic Mechanics, Math. Appl. 35, Reidel, Dordrecht, 1987.
26. Mishchenko, A. S.: Integration of geodesic flows on symmetric spaces, Mat. Zametki 31(2) (1982), 257-262 [in Russian]. English translation: Math. Notes 31(1-2) (1982), 132-134.
27. Mishchenko, A. S.: Integration of geodesic flows on symmetric spaces, Mat. Zametki 31(2) (1982), 257-262 [in Russian]. English translation: Math. Notes 31(1-2) (1982), 132-134.
28. Mishchenko, A. S. and Fomenko, A. T.: Euler equations on finite-dimensional Lie groups, Izv. Akad. Nauk SSSR, Ser. Mat. 42(2) (1978), 396-415 [in Russian]. English translation: Math. USSR-Izv. 12(2) (1978), 371-389.
29. Mishchenko, A. S. and Fomenko, A. T.: Euler equations on finite-dimensional Lie groups, Izv. Akad. Nauk SSSR, Ser. Mat. 42(2) (1978), 396-415 [in Russian]. English translation: Math. USSR-Izv. 12(2) (1978), 371-389.
30. Mishchenko, A. S. and Fomenko, A. T.: Generalized Liouville method of integration of Hamiltonian systems, Funktsional. Anal. i Prilozhen. 12(2) (1978), 46-56, [in Russian]. English translation: Functional Anal. Appl. 12 (1978), 113-121.
31. Mishchenko, A. S. and Fomenko, A. T.: Generalized Liouville method of integration of Hamiltonian systems, Funktsional. Anal. i Prilozhen. 12(2) (1978), 46-56, [in Russian]. English translation: Functional Anal. Appl. 12 (1978), 113-121.
32. Mikityuk, I. V.: Homogeneous spaces with integrable G-invariant Hamiltonian flows, Izv. Akad. Nauk SSSR, Ser. Mar. 47(6) (1983), 1248-1262 [in Russian]. English translation in Math. USSR-Izv. (1983).
33. Mikityuk, I. V.: Homogeneous spaces with integrable G-invariant Hamiltonian flows, Izv. Akad. Nauk SSSR, Ser. Mar. 47(6) (1983), 1248-1262 [in Russian]. English translation in Math. USSR-Izv. (1983).
34. Mikityuk, I. V.: On the integrability of invariant Hamiltonian systems with homogeneous configuration spaces, Mat. Sb., 129(4) (1986), 514-534 [in Russian]. English translation: Math. USSR-Sb. 57(2) (1987), 527-547.
35. Mikityuk, I. V.: On the integrability of invariant Hamiltonian systems with homogeneous configuration spaces, Mat. Sb., 129(4) (1986), 514-534 [in Russian]. English translation: Math. USSR-Sb. 57(2) (1987), 527-547.
36. Nekhoroshev, N. N.: Action-angle variables and their generalization, Trudy Moskov Mat. Obshch., 26 (1972), 181-198 [in Russian]. English translation: Trans. Moscow Math. Soc. 26 (1972), 180-198.
37. Nekhoroshev, N. N.: Action-angle variables and their generalization, Trudy Moskov Mat. Obshch., 26 (1972), 181-198 [in Russian]. English translation: Trans. Moscow Math. Soc. 26 (1972), 180-198.
38. Onishchik, A. L., Vinberg, E. B. and Gorbatsevich, V. V.: Foundations of Lie Theory and Lie Transformation Groups, Springer, New York, 1993.
39. Paternain, G. P. and Spatzier, R. J.: New examples of manifolds with completely integrable geodesic flows, Adv. Math. 108 (1994), 346-366.
40. Schwarz, G. W.: Smooth functions invariant under the action of a compact Lie group, Topology 14(1) (1975), 63-68.
41. Taimanov, I. A.: Topological obstructions to integrability of geodesic flows on non-simply-connected manifolds, Izv. Akad. Nauk SSSR, Ser. Mat. 51(2) (1987), 429-435 [in Russian]. English translation: Math. USSR-Izv. 30(2) (1988), 403-409.
42. Taimanov, I. A.: Topological obstructions to integrability of geodesic flows on non-simply-connected manifolds, Izv. Akad. Nauk SSSR, Ser. Mat. 51(2) (1987), 429-435 [in Russian]. English translation: Math. USSR-Izv. 30(2) (1988), 403-409.
43. Taimanov, I. A.: On the topological properties of integrable geodesic flows, Mat. Zametki 44(3) (1988), 283-284 [in Russian]. English translation: Math. Notes 44 (1988).
44. Taimanov, I. A.: On the topological properties of integrable geodesic flows, Mat. Zametki 44(3) (1988), 283-284 [in Russian]. English translation: Math. Notes 44 (1988).
45. Thimm, A.: Integrable geodesic flows on homogeneous spaces, Ergodic Theory Dynamical Systems 1 (1981), 495-517.
46. Trofimov, V. V. and Fomenko, A. T.: Geometric and algebraic mechanisms of the integrability of Hamiltonian systems on homogeneous spaces and Lie algebras, in V. I. Arnold and S. P. Novikov (eds), Dynamical Systems VII, Springer, New York, 1994, pp. 261-333.
47. Trofimov, V. V. and Fomenko, A. T.: Algebra and Geometry of Integrable Hamiltonian Differential Equations, Faktorial, Moscow, 1995 [in Russian].
48. Wadsley, A. W.: Geodesic foliations by circles, J. Differential Geom. 10 (1975), 541-549.