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Integrability of invariant geodesic flows on n-symmetric spaces

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Abstract

In this article, by modifying the argument shift method, we prove Liouville integrability of geodesic flows of normal metrics (invariant Einstein metrics) on the Ledger–Obata n-symmetric spaces K n/diag(K), where K is a semisimple (respectively, simple) compact Lie group.

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References

  1. Besse A.: Einstein Manifolds: A Series of Modern Surveys in Mathematics. Springer, Berlin (1987)

    Google Scholar 

  2. Bolsinov, A.V.: Compatible Poisson brackets on Lie algebras and the completeness of families of functions in involution, Izv. Acad. Nauk SSSR, Ser. Matem. 55(1), 68–92 (1991) (Russian); English transl.: Math. USSR-Izv. 38(1), 69–90 (1992)

  3. Bolsinov, A.V., Jovanović, B.: Integrable geodesic flows on homogeneous spaces. Matem. Sbornik 192(7), 21–40 (2001) (Russian); English transl.: Sb. Mat. 192(7–8), 951–969 (2001)

    Google Scholar 

  4. Bolsinov A.V., Jovanović B.: Non-commutative integrability, moment map and geodesic flows. Ann. Glob. Anal. Geom. 23(4), 305–322 (2003) arXiv: math-ph/0109031

    Article  MATH  Google Scholar 

  5. Bolsinov A.V., Jovanović B.: Complete involutive algebras of functions on cotangent bundles of homogeneous spaces. Math. Z. 246(1–2), 213–236 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  6. Brailov, A.V.: Construction of complete integrable geodesic flows on compact symmetric spaces. Izv. Acad. Nauk SSSR, Ser. Matem. 50(2), 661–674 (1986) (Russian); English transl.: Math. USSR-Izv. 50(4), 19–31 (1986)

  7. Butler L.: Integrable geodesic flows with wild first integrals: the case of two-step nilmanifolds. Ergodic Theory Dynam. Systems 23(3), 771–797 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  8. Dragović V., Gajić B., Jovanović B.: Singular Manakov flows and geodesic flows of homogeneous spaces of SO(n). Transfom. Groups 14(3), 513–530 (2009) arXiv: 0901.2444

    Article  MATH  Google Scholar 

  9. Ledger A.J., Obata M.: Affine and Riemannian s-manifolds. J. Differential Geom. 2, 451–459 (1968)

    MATH  MathSciNet  Google Scholar 

  10. Magazev, A.A., Shirokov, I.V.: Integration of geodesic flows on homogeneous spaces. The case of a wild Lie group. Teoret. Mat. Fiz. 136(3), 365–379 (2003) (Russian); English transl.: Theoret. Math. Phys. 136(3), 1212–1224 (2003)

  11. Mishchenko, A.S.: Integration of geodesic flows on symmetric spaces. Mat. Zametki 31(2), 257–262 (1982) (Russian); English transl.: Math. Notes 31(1–2), 132–134 (1982)

    Google Scholar 

  12. Mishchenko, A.S., Fomenko, A.T.: Euler equations on finite-dimensional Lie groups. Izv. Acad. Nauk SSSR Ser. Matem. 42(2), 396–415 (1978) (Russian); English transl.: Math. USSR Izv. 12(2), 371–389 (1978)

  13. Mishchenko, A.S., Fomenko, A.T.: Generalized Liouville method of integration of Hamiltonian systems. Funkts. Anal. Prilozh. 12(2), 46–56 (1978) (Russian); English transl.: Funct. Anal. Appl. 12, 113–121 (1978)

  14. Mikityuk, I.V.: On the integrability of invariant Hamiltonian systems with homogeneous configuration spaces. Matem. Sbornik, 129(171)(4), 514–534 (1986) (Russian); English transl.: Math. USSR Sbornik., 57(2), 527–547 (1987)

  15. Mykytyuk I.V., Panasyuk A.: Bi-Poisson structures and integrability of geodesic flows on homogeneous spaces. Transform. Groups 9(3), 289–308 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  16. Nekhoroshev, N.N.: Action-angle variables and their generalization. Tr. Mosk. Mat. Obs. 26, 181–198 (1972) (Russian); English transl.: Trans. Moscow Math. Soc. 26, 180–198 (1972)

  17. Nikonorov, Yu.G.: Invariant Einstein metrics on the Ledger–Obata Spaces. Algebra i Analiz. 14(3), (2002) (Russian); English transl.: St. Petersburg Math. J. 14(3), 487–497 (2002)

  18. Panasyuk, A.: Projection of Jordan bi-Poisson structures that are Kronecker, diagonal actions and the classical Gaudin systems. J. Geom. Phys. 47, 379–397 (2003); Erratum: J. Geom. Phys. 49, 116–117 (2004)

  19. Panyushev D.I., Yakimova O.S.: The argument shift method and maximal commutative subalgebras of Poisson algebras. Math. Res. Lett. 15(2), 239–249 (2008) arXiv: math/0702583

    MATH  MathSciNet  Google Scholar 

  20. Sadetov S.T.: A proof of the Mishchenko-Fomenko conjecture (1981). Dokl. Akad. Nauk. 397(6), 751–754 (2004) (Russian)

    MathSciNet  Google Scholar 

  21. Thimm A.: Integrable geodesic flows on homogeneous spaces. Ergodic Theory Dynam. Systems 1, 495–517 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  22. Wang M., Ziller W.: On normal homogeneous Einstein manifolds. Ann. Sci. École Norm. Sup. 18(4), 563–633 (1985)

    MATH  MathSciNet  Google Scholar 

  23. Zung N.T.: Torus actions and integrable systems. In: Bolsinov, A.V., Fomenko, A.T., Oshemkov, A.A. (eds) Topological Methods in the Theory of Integrable Systems, pp. 289–328. Cambridge Scientific Publishers, Cambridge (2006) arXiv: math.DS/0407455

    Google Scholar 

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  1. Mathematical Institute, Serbian Academy of Sciences and Arts, Kneza Mihaila 36, 11000, Belgrade, Serbia

    Božidar Jovanović

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  1. Božidar Jovanović
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Correspondence to Božidar Jovanović.

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Jovanović, B. Integrability of invariant geodesic flows on n-symmetric spaces. Ann Glob Anal Geom 38, 305–316 (2010). https://doi.org/10.1007/s10455-010-9216-2

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  • DOI: https://doi.org/10.1007/s10455-010-9216-2

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