Advertisement

Journal of Mathematical Sciences

, Volume 146, Issue 6, pp 6213–6312 | Cite as

Diffeomorphism groups of compact manifolds

  • N. K. Smolentsev
Article

Keywords

Manifold Euler Equation Sectional Curvature Symplectic Form Symplectic Manifold 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    K. Abe and K. Fukui, “On commutators of equivariant diffeomorphisms,” Proc. Jpn. Acad., 54, 52–54 (1978).zbMATHMathSciNetGoogle Scholar
  2. 2.
    K. Abe and K. Fukui, “On the structure of automorphisms of manifolds,” in: Proc. Int. Conf. on Geometry, Integrability, and Quantization, Varna, Bulgaria, September 1–10, 1999 (I. M. Mladenov et al., eds.), Coral Press Scientific Publ., Sofia (2000), pp. 7–16.Google Scholar
  3. 3.
    K. Abe, “On the homotopy type of groups of equivariant diffeomorphisms,” Publ. RIMS Kyoto Univ., 16, 601–626 (1980).zbMATHGoogle Scholar
  4. 4.
    R. Abraham, Lectures of Smale on Differential Topology, Mimeographed notes, Columbia Univ., New York (1962).Google Scholar
  5. 5.
    R. Abraham and J. Marsden, Foundations of Mechanics, Benjamin, New York (1967).zbMATHGoogle Scholar
  6. 6.
    M. Adams, T. Ratiu, and R. Schmid, “A Lie group structure for pseudodifferential operators,” Math. Ann., 273, No. 4, 529–551 (1986).zbMATHMathSciNetGoogle Scholar
  7. 7.
    M. Adams, T. Ratiu, and R. Schmid, “A Lie group structure for Fourier integral operators,” Math. Ann., 276, No. 1, 19–41 (1986).zbMATHMathSciNetGoogle Scholar
  8. 8.
    G. D’Ambra and M. Gromov, “Lecture on transformation groups: Geometry and dynamics,” Surv. Differ. Geom., 1, 19–111 (1991).MathSciNetGoogle Scholar
  9. 9.
    P. Antonelli, D. Burghelea, and P. Kahn, “The non-infinite homotopy type of some diffeomorphism groups,” Topology, 11, 1–49 (1972).zbMATHMathSciNetGoogle Scholar
  10. 10.
    T. A. Arakelyan and G. K. Savvidy, “Geometry of a group of area-preserving diffeomorphisms,” Phys. Lett. B, 223, No. 1, 41–46 (1989).MathSciNetGoogle Scholar
  11. 11.
    V. I. Arnold, “Variational principle for three-dimensional stationary flows of the ideal fluid,” Prikl. Mat. Mekh., 29, No. 5, 846–851 (1965).Google Scholar
  12. 12.
    V. I. Arnold, “Sur la topologie des écoulements stationnaires des fluides parfaits,” C. R. Acad. Sci. Paris, 261, 117–120 (1965).Google Scholar
  13. 13.
    V. I. Arnold, “Sur la geometrie differentielle des groupes de Lie de dimension infinite et ses applications a l’hidrodynamique des fluides parfaits,” Ann. Inst. Fourier, 16, No. 1, 319–361 (1966).MathSciNetGoogle Scholar
  14. 14.
    V. I. Arnold, “Hamiltonian property of the Euler equations of the rigid body and ideal fluid dynamics,” Usp. Mat. Nauk., 24, No. 3, 225–226 (1969).Google Scholar
  15. 15.
    V. I. Arnold, Mathematical Methods of Classical Mechanics [in Russian], Nauka, Moscow (1974).Google Scholar
  16. 16.
    V. I. Arnold and A. B. Givental, “Symplectic geometry,” in: Progress in Science and Technology, Series on Contemporary Problems in Mathematics, Fundamental Directions, Dynamical System-4 [in Russian], All-Union Institute for Scientific and Technical Information, USSR Academy of Sciences, Moscow (1985), pp. 7–139.Google Scholar
  17. 17.
    V. I. Arnold, “The asymptotic Hopf invariant and its applications,” Select. Math. Sov., 5, 327–345 (1986).zbMATHGoogle Scholar
  18. 18.
    V. I. Arnold, “The asymptotic Hopf invariant and its applications,” Select. Math. Sov., 5, No. 4, 327–345 (1986).zbMATHGoogle Scholar
  19. 19.
    V. I. Arnold and B. Khesin Topological Methods in Hydrodynamics, Springer Verlag, New York (1998).zbMATHGoogle Scholar
  20. 20.
    M. F. Atiyah, V. K. Patodi, and I. M. Singer, “Spectral asymmetry and Riemannian geometry, I,” Math. Proc. Cambridge Phil. Soc., 77, 43–69 (1975).zbMATHMathSciNetGoogle Scholar
  21. 21.
    M. F. Atiyah, V. K. Patodi, and I. M. Singer, “Spectral asymmetry and Riemannian geometry, II,” Math. Proc. Cambridge Phil. Soc., 78, 405–432 (1975).zbMATHMathSciNetGoogle Scholar
  22. 22.
    V. I. Averbukh and O. G. Smolyanov, “Differentiation theory in linear topological spaces,” Usp. Math. Nauk, 22, No. 6, 201–260 (1967).Google Scholar
  23. 23.
    V. I. Averbukh and O. G. Smolyanov, “Various definitions of the derivative in linear topological spaces,” Usp. Mat. Nauk, 23, No. 4, 67–116 (1968).Google Scholar
  24. 24.
    A. Banyaga, “On the group of equivariant diffeomorphisms,” Topology, 16, 279–283 (1977).zbMATHMathSciNetGoogle Scholar
  25. 25.
    A. Banyaga, “Sur la structure du groupe des diffeomorphismes qui preservent une forme symplectique,” Comment. Math. Helvet., 53, 174–227 (1978).zbMATHMathSciNetGoogle Scholar
  26. 26.
    A. Banyaga, “The group of diffeomorphisms preserving a regular contact form,” Monogr. Enseign. Math., 26, 47–53 (1978).MathSciNetGoogle Scholar
  27. 27.
    A. Banyaga and J. Pulido, “On the group of contact diffeomorphisms of ℝ2n+1,” Bort. Soc. Matem. Mexicana, 23, No. 2, 43–47 (1978).zbMATHMathSciNetGoogle Scholar
  28. 28.
    A. Banyaga, “On fixed points of symplectic maps,” Invent Math., 56, 215–229 (1980).MathSciNetGoogle Scholar
  29. 29.
    A. Banyaga, “Sur la cohomologie du groupe des diffeomorphismes,” C. R. Acad.Sci. Paris, Ser. I, 294, 625–627 (1982).zbMATHMathSciNetGoogle Scholar
  30. 30.
    A. Banyaga, “On isomorphic classical diffeomorphism groups, II,” J. Differ. Geom., 28, No. 1, 23–35 (1988).zbMATHMathSciNetGoogle Scholar
  31. 31.
    A. Banyaga, “Sur la groupe des diffeomorphismes symplectiques,” Lect. Notes. Math., 484, 50–56 (1975).MathSciNetGoogle Scholar
  32. 32.
    A. Banyaga, The Structure of Classical Diffeomorphisms Groups, Kluwer Academic Publ., Amsterdam (1997).Google Scholar
  33. 33.
    D. Bao, J. Lafontaine, T. Ratiu, “On a nonlinear equation related to the geometry of the diffeomorphism groups,” Pac. J. Math., 158, 223–242 (1993).MathSciNetGoogle Scholar
  34. 34.
    Yu. S. Baranov and Yu. E. Gliklikh, “A note on the regularity of solutions of the Euler equations of hydrodynamics,” Usp. Mat. Nauk, 36, No. 5, 163–164 (1981).zbMATHMathSciNetGoogle Scholar
  35. 35.
    Yu. S. Baranov and Yu. E. Gliklikh, “One mechanical connection of the volume-preserving diffeomorphism group,” Funkts. Anal. Prilozh., 22, No. 2, 61–62 (1988).MathSciNetGoogle Scholar
  36. 36.
    Yu. S. Baranov and Yu. E. Gliklikh, “Some applications of the geometry of infinite-dimensional manifolds in hydrodynamics,” in: Geometry and Topology in Global Nonlinear Problems [in Russian], VGU, Voronezh (1984), pp. 142–158.Google Scholar
  37. 37.
    M. Benaim and J.-M. Gambaudo, “Metric properties of the group of area preserving diffeomorphisms,” Trans. Amer. Math. Soc., 353, No. 11, 4661–4672 (2001).zbMATHMathSciNetGoogle Scholar
  38. 38.
    D. Behheken, “Elliptic problems, Riemannian surfaces, and (M. Gromov) symplectic structures,” in: Mathematical Analysis and Geometry, Series “News in Foreign Science” [in Russian], 45, Mir, Moscow (1990), pp. 183–206.Google Scholar
  39. 39.
    M. Berger and D. Ebin, “Some decompositions of the space of symmetric tensors on a Riemannian manifold,” J. Differ. Geom., 3, No. 3, 379–392 (1969).zbMATHMathSciNetGoogle Scholar
  40. 40.
    A. Besse, Four-Dimensional Riemannian Geometry [Russian translation], Mir, Moscow (1985).Google Scholar
  41. 41.
    A. Besse, Einstein Manifolds, Vols. 1, 2 [Russian translation], Mir, Moscow (1990).zbMATHGoogle Scholar
  42. 42.
    M. Bialy and L. Polterovich, “Hamiltonian diffeomorphisms and Lagrangian distribution,” Geom. Funct. Anal., 2, No. 2, 173–210 (1992).zbMATHMathSciNetGoogle Scholar
  43. 43.
    J. M. Bismut and J. Lott, “Flat vector bundles, direct images, and higher real analytic torsion,” J. Amer. Math. Soc., 8, 291–363 (1995).zbMATHMathSciNetGoogle Scholar
  44. 44.
    L. Bitam, FrSur la type d’homotopie des groupes classiques de diffeomorphismes, Thèse Doct. 3ème cycle Math. Pures Univ. Sci. et Med. Grenoble (1984).Google Scholar
  45. 45.
    D. E. Blair, Contact Manifolds in Riemannian Geometry, Lect. Notes Math., 509, Springer-Verlag (1976).Google Scholar
  46. 46.
    W. M. Boothby, “The transitivity of the automorphisms of certain geometric structures,” Trans. Amer. Math. Soc., 137, 93–100 (1969).zbMATHMathSciNetGoogle Scholar
  47. 47.
    R. Bott, “On the characteristic classes of group of diffeomorphisms,” Monogr. Enseign. Math., 26, 63–74 (1978).Google Scholar
  48. 48.
    S. Bouarroudj and V. Yu. Ovsienko, “Three cocycles on Di.(S 1) generalizing the Schwarzian derivative,” Int. Math. Res. Notices, 1, 25–39 (1998).MathSciNetGoogle Scholar
  49. 49.
    R. Brooks, “Volumes and characteristic classes of foliations,” Topology, 18, 295–304 (1979).zbMATHMathSciNetGoogle Scholar
  50. 50.
    U. Bunke, Higher analytic torsion and cohomology of diffeomorphism groups, E-print dg-ga/9712001 (1997), http://xxx.lanl.gov.Google Scholar
  51. 51.
    U. Bunke, “Higher analytic torsion of sphere bundles and continuous cohomology of Diff(S 2n−1),” E-print math.DG/9802100 (1998), http://xxx.lanl.gov.Google Scholar
  52. 52.
    D. Burghelea, “On the homotopy type of Diff(M n) and connected problems,” Colloq. Int. CNRS, 7, No. 210, 3–17 (1973).MathSciNetGoogle Scholar
  53. 53.
    E. Calabi, “On the group of automorphisms of a symplectic manifold,” in: Problems in Analysis. Symp. in Honor of S. Bochner, Princeton Univ. Press (1970), pp. 1–26.Google Scholar
  54. 54.
    J. Cheeger and J. Simons, “Differential characters and geometric invariants,” Lect. Notes Math., 1167, 50–80 (1985).MathSciNetGoogle Scholar
  55. 55.
    S. Chern and J. Simons, “Characteristic forms and geometric invariants,” Ann. Math., 99, No. 1, 48–69 (1974).MathSciNetGoogle Scholar
  56. 56.
    P. R. Chernnoff, “Irreducible representations of infinite-dimensional transformation groups and Lie algebras,” Bull. Amer. Math. Soc., 13, No. 1, 46–48 (1985).MathSciNetGoogle Scholar
  57. 57.
    Y. M. Choi, K. S. Soh, and J. H. Yoon, “Gravitations as gauge theory of diffeomorphism group,” Phys. Rev. D, 91, 1–10 1991.Google Scholar
  58. 58.
    A. Constantin and B. Kolev, “On the geometric approach to the motion inertial mechanical systems,” J. Phys. A, 35, R51–R79 (2002).zbMATHMathSciNetGoogle Scholar
  59. 59.
    A. Constantin and B. Kolev, “Geodesic flow on the diffeomorphism group of the circle,” Comment. Math. Helv., 78, 787–804 (2003).zbMATHMathSciNetGoogle Scholar
  60. 60.
    B. Dai and H.-Y. Wang, “A note on diffeomorphism groups of closed manifolds,” Ann. Global Anal. Geom., 21, No. 2, 135–140 (2002).zbMATHMathSciNetGoogle Scholar
  61. 61.
    A. A. Dezin, “Invariant forms and some structure properties of the Euler equations of hydrodynamics,” Z. Anal. Anwend., 2, 401–409 (1983).zbMATHMathSciNetGoogle Scholar
  62. 62.
    S. K. Donaldson, “Moment maps and diffeomorphisms,” Asian J. Math., 3, No. 1, 1–16 (1999).zbMATHMathSciNetGoogle Scholar
  63. 63.
    W. G. Dwyer and R. H. Szczarba, “Sur l’homotopie des groupes de diffeomorphismes,” C. R. Acad. Sci. Paris, Ser. A, 289, 417–419 (1979).zbMATHMathSciNetGoogle Scholar
  64. 64.
    C. J. Earle and J. Eells, “The diffeomorphism group of a compact Riemannian surface,” Bull. Amer. Math. Soc., 73, No. 4, 557–559 (1967).MathSciNetGoogle Scholar
  65. 65.
    C. J. Earle and J. Eells, “A fibre bundle description of Teichmuller theory,” J. Differ. Geom., 3, 19–43 (1969).zbMATHMathSciNetGoogle Scholar
  66. 66.
    D. Ebin, “The manifold of Riemannian metrics,” Proc. Symp. Pure Math., 15, 11–40 (1970).MathSciNetGoogle Scholar
  67. 67.
    D. Ebin, “Integrability of perfect fluid motion,” Commun. Pure Appl. Math., 36, No. 1, 37–54 (1983).zbMATHMathSciNetGoogle Scholar
  68. 68.
    D. Ebin and J. Marsden, “Groups of diffeomorphisms and the motion of an incompressible fluid,” Ann. Math., 92, No. 1, 102–163 (1970).MathSciNetGoogle Scholar
  69. 69.
    J. Eells, “On the geometry of function spaces,” in: Symp. Topology Algebra, Mexico (1958), pp. 303–307.Google Scholar
  70. 70.
    J. Eells, “On submanifolds of certain function spaces,” Proc. Natl. Acad. Sci., 45, No. 10, 1520–1522 (1959).zbMATHMathSciNetGoogle Scholar
  71. 71.
    J. Eells, “Alexander-Pontryagin duality in function spaces,” Proc. Symp. Pure Math., 3, 109–129 (1961).MathSciNetGoogle Scholar
  72. 72.
    J. Eells, “A setting for global analysis,” Bull. Amer. Math. Soc., 72, 751–787 (1966).MathSciNetGoogle Scholar
  73. 73.
    J. Eichhorn, “The manifold structure of maps between open manifolds,” Ann. Global Anal. Geom., 11, 253–300 (1993).zbMATHMathSciNetGoogle Scholar
  74. 74.
    J. Eichhorn, “Gauge theory on open manifolds of bounded geometry,” Int. J. Mod. Phys., 7, 3927–3977 (1993).MathSciNetGoogle Scholar
  75. 75.
    J. Eichhorn, “Spaces of Riemannian metrics on open manifolds,” Results Math., 27, 256–283 (1995).zbMATHMathSciNetGoogle Scholar
  76. 76.
    J. Eichhorn and R. Schmid, “Form preserving diffeomorphisms on open manifolds,” Ann. Global Anal. Geom., 14, 147–176 (1996).zbMATHMathSciNetGoogle Scholar
  77. 77.
    J. Eichhorn, “Diffeomorphism groups on noncompact manifolds,” Zap. Nauch. Semin. POMI, 234, 41–64 (1996).Google Scholar
  78. 78.
    J. Eichhorn and J. Fricke, “The module structure theorem for Sobolev spaces on open manifolds,” Math. Nachr., 184, 35–47 (1998).MathSciNetGoogle Scholar
  79. 79.
    J. Eichhorn, “Poincaré’s theorem and Teichmuller theory for open manifolds,” Asian J. Math., 2, No. 2, 355–404 (1998).zbMATHMathSciNetGoogle Scholar
  80. 80.
    J. Eichhorn, “A classification approach for open manifolds,” Zap. Nauchn. Semin. POMI, 267, 9–45 (2000).Google Scholar
  81. 81.
    Ya. Eliashberg and L. Polterovich, “Bi-invariant metrics on the group of Hamiltinian diffeomorphisms,” Int. J. Math., 4, No. 5, 727–738 (1993).zbMATHMathSciNetGoogle Scholar
  82. 82.
    Ya. Eliashberg and T. Ratiu, “The diameter of the symplectomorphism group is infinite,” Invent. Math., 103, No. 2, 327–340 (1991).zbMATHMathSciNetGoogle Scholar
  83. 83.
    H. Eliasson, “On the geometry of manifold of maps,” J. Differ. Geom., 1, 169–194 (1967).zbMATHMathSciNetGoogle Scholar
  84. 84.
    D. B. A. Epstein, “The simplicity of certain groups of homeomorphisms,” Compos. Math., 22, 165–173 (1970).zbMATHGoogle Scholar
  85. 85.
    D. B. A. Epstein, “Commutators of C diffeomorphisms,” Comment. Math. Helv., 59, 111–122 (1984).zbMATHMathSciNetGoogle Scholar
  86. 86.
    J. Etnyre and R. Ghrist, Contact topology and hydrodynamics, II: Solid tori, E-print math.SG/9907112 (1999), http://xxx.lanl.gov.Google Scholar
  87. 87.
    J. Etnyre and R. Ghrist, Contact topology and hydrodynamics, III: Knotted flowlines, E-print math-ph/9906021 (1999), http://xxx.lanl.gov.Google Scholar
  88. 88.
    J. Etnyre and R. Ghrist, An index for closed orbits in Beltrami fields, E-print math.DS/0101095 (2001), http://xxx.lanl.gov.Google Scholar
  89. 89.
    R. P. Filipkewicz, “Isomorphisms between diffeomorphism groups,” Ergodic Theor. Dynam. Syst., 2, 159–171 (1982).Google Scholar
  90. 90.
    A. Fischer and A. Tromba, “On a purely ’Riemannian’ proof of the structure and dimension of the unramiffed moduli space of a compact Riemannian surface,” Math. Ann., 267, 311–345 (1984).zbMATHMathSciNetGoogle Scholar
  91. 91.
    E. G. Floratos and J. Iliopoulos, “A note on the classical symmetries of the closed bosonic membranes,” Phys. Lett. B, 201, No. 2, 237–240 (1988).MathSciNetGoogle Scholar
  92. 92.
    D. B. Fuks, “Cohomologies of infinite-dimensional Lie algebras and characteristic classes of foliations,” in: Progress in Science and Technology, Series on Contemporary Problems in Mathematics [in Russian], 10, All-Union Institute for Scientific and Technical Information, USSR Academy of Sciences, Moscow (1978), pp. 179–285.Google Scholar
  93. 93.
    D. B. Fuks, Cohomologies of Infinite-Dimensional Lie Algebras [in Russian], Nauka, Moscow (1984).Google Scholar
  94. 94.
    B. L. Feigin and D. B. Fuks, “Cohomologies of Lie groups and algebras,” in: Progress in Science and Technology, Series on Contemporary Problems in Mathematics, Fundamental Directions [in Russian], 21, All-Union Institute for Scientific and Technical Information, USSR Academy of Sciences, Moscow (1988), pp. 121–209.Google Scholar
  95. 95.
    K. Fukui and S. Ushiki, “On the homotopy type of FDiff \((S^3 ,\mathcal{F}_R )\),” J. Math. Kyoto Univ., 15, No. 1, 201–210 (1975).zbMATHMathSciNetGoogle Scholar
  96. 96.
    K. Fukui, “Homologies of the group of Diff(ℝn, 0),” J. Math. Kyoto Univ., 20, 475–487 (1980).zbMATHMathSciNetGoogle Scholar
  97. 97.
    I. M. Gel’fand and D. B. Fuks, “Cohomologies of Lie algebras of vector fields on the circle,” Functs. Anal. Prilozh., 2, No. 4, 92–93 (1968).zbMATHMathSciNetGoogle Scholar
  98. 98.
    V. L. Ginzburg, “Some remarks on symplectic actions of compact groups,” Math. Z., 210, 625–640 (1992).zbMATHMathSciNetGoogle Scholar
  99. 99.
    Yu. E. Gliklikh, Analysis on Riemannian Manifolds and Problems of Mathematical Physics [in Russian], VGU, Voronezh (1989).zbMATHGoogle Scholar
  100. 100.
    K. Godbillon, Differential Geometry and Analytic Mechanics [Russian translation], Mir, Moscow (1973).Google Scholar
  101. 101.
    M. Golubitsky and V. W. Guillemin, Stable Mappings and Their Singularities, Grad. Texts Math., 14, Springer-Verlag (1973).Google Scholar
  102. 102.
    J. Grabowski, “Free subgroups of diffeomorphisms groups,” Fundam. Math., 131, No. 2, 103–121 (1988).zbMATHMathSciNetGoogle Scholar
  103. 103.
    A. Gramain, “Le type d’homotopie du groupe des diffeomorphismes d’une surface compacte,” Ann. Sci. Ecole Norm. Super., 6, No. 1, 53–66 (1973).zbMATHMathSciNetGoogle Scholar
  104. 104.
    R. E. Greene and K. Shiohama, “Diffeomorphisms and volume-preserving embeddings of noncompact manifolds,” Trans. Amer. Math. Soc., 255, 403–414 (1979).zbMATHMathSciNetGoogle Scholar
  105. 105.
    M. Gromov, “Pseudoholomorphic curves in symplectic manifolds,” Invent. Math., 82, No. 2, 307–347 (1985).zbMATHGoogle Scholar
  106. 106.
    M. Gromov, Partial Differential Relations [Russian translation], Mir, Moscow (1990).Google Scholar
  107. 107.
    M. Gromov, “Flexible and rigid symplecti topology,” in: Berkeley International Congress of Mathematicians, 1996, Overview Reports [in Russian], Mir, Moscow (1996), pp. 139–163.Google Scholar
  108. 108.
    D. Gromol, W. Klingenberg, and W. Meyer, Riemannian Geometry in the Large [Russian translation], Mir, Moscow (1971).Google Scholar
  109. 109.
    V. W. Guillemin, “Infinite-dimensional primitive Lie algebras,” J. Differ. Geom., 4, 257–282 (1970).zbMATHMathSciNetGoogle Scholar
  110. 110.
    S. Haller and T. Rybicki, On the perfectness of nontransitive groups of diffeomorphisms, E-print math.DG/9902095 (1999), http://xxx.lanl.gov.Google Scholar
  111. 111.
    R. S. Hamilton, “The inverse function theorem of Nash and Moser,” Bull. Amer. Math. Soc., 7, No. 1, 65–222 (1982).zbMATHMathSciNetGoogle Scholar
  112. 112.
    D. Hart, “On the smoothness of generators,” Topology, 22, No. 3, 357–363 (1983).zbMATHMathSciNetGoogle Scholar
  113. 113.
    Y. Hatakeyama, “Some notes on the groups of automorphisms of contact and symplectic structures,” Tohoku Math. J., 18, 338–347 (1966).zbMATHMathSciNetGoogle Scholar
  114. 114.
    A. Hatcher, “A proof of the Smale conjecture Diff(S 3) ≅ O(4),” Ann. Math., 117, 553–607 (1983).MathSciNetGoogle Scholar
  115. 115.
    A. Hatcher and D. McCullough, Finiteness of classifying spaces of relative diffeomorphism groups of 3-manifolds, E-print math.GT/9712260 (1997), http://xxx.lanl.gov.Google Scholar
  116. 116.
    Y. Hattori, “Ideal magnetohydrodynamics and passive scalar motion as geodesics on semidirect product groups,” J. Phys. A: Math. Gen., 27, L21–L25 (1994).MathSciNetGoogle Scholar
  117. 117.
    M. R. Herman, “Simplicite du groupe des diffeomorphismes de classe C , isotopes al’identite, du tore de dimension n,” C. R. Acad. Sci. Paris, Ser. A, 273, 232–234 (1971).zbMATHMathSciNetGoogle Scholar
  118. 118.
    M. R. Herman, “Sur la groupe des diffeomorphismes du tore,” Ann. Inst. Fourier, 23, No. 2, 75–86 (1973).zbMATHMathSciNetGoogle Scholar
  119. 119.
    M. R. Herman, “Sur la conjugaison differentiable des diffeomorphismes du cercle a des rotations,” Publ. Math. IHES, 49, 5–234 (1979).zbMATHMathSciNetGoogle Scholar
  120. 120.
    M. W. Hirsch, Differential Topology, Grad. Texts Math., 33, Springer-Verlag (1976).Google Scholar
  121. 121.
    D. D. Holm, J. E. Marsden, and T. S. Ratiu, “Euler-Poincaré models of ideal fluids with nonlinear dispersion,” Phys. Rev. Lett., 349, 4173–4277 (1998).Google Scholar
  122. 122.
    D. D. Holm, J. E. Marsden, and T. S. Ratiu, “Euler-Poincaré equations and semidirect products with applications to continuum theories,” Adv. Math., 137, 1–81 (1998).zbMATHMathSciNetGoogle Scholar
  123. 123.
    H. Hofer, “Estimates for the energy of a symplectic map,” Comment. Math. Helv., 68, No. 1, 48–72 (1993).zbMATHMathSciNetGoogle Scholar
  124. 124.
    H. Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics, Birkhauser, Basel-Boston-Berlin (1994).zbMATHGoogle Scholar
  125. 125.
    J. Hoppe, “DiffA T 2 and the curvature of some infinite dimensional manifolds,” Phys. Lett. B, 215, No. 4, 706–710 (1988).MathSciNetGoogle Scholar
  126. 126.
    D. Husemoller, Fibre Bundles, McGraw-Hill (1966).Google Scholar
  127. 127.
    R. S. Ismagilov, “On unitary representations of diffeomorphism groups of the circle,” Functs. Anal. Prilozh., 5, No. 3, 45–54 (1971).MathSciNetGoogle Scholar
  128. 128.
    R. S. Ismagilov, “On unitary representations of diffeomorphisms groups of a compact manifold,” Izv. Akad. Nauk SSSR, Ser. Mat., 36, No. 1, 180–208 (1972).zbMATHMathSciNetGoogle Scholar
  129. 129.
    R. S. Ismagilov, “On unitary representations of diffeomorphism groups of the space ℝn, n ≥ 2,” Funkts. Anal. Prilozh., 9, 71–72 (1975).MathSciNetGoogle Scholar
  130. 130.
    R. S. Ismagilov, “On unitary representation of diffeomorphism groups of the space C 0 (X,G), G = XU(2),” Mat. Sb., 100, No. 1, 117–131 (1976).MathSciNetGoogle Scholar
  131. 131.
    R. S. Ismagilov, “Unitary representations of the measure-preserving diffeomorphism group,” Funkts. Anal. Prilozh., 1, No. 3, 80–81 (1977).Google Scholar
  132. 132.
    R. S. Ismagilov, “Inductive limits of the area-preserving diffeomorphism groups,” Funkts. Anal. Prilozh., 37, No. 3, 36–50 (2003).MathSciNetGoogle Scholar
  133. 133.
    J. Kedra, “Remarks on the flux groups,” Math. Res. Lett., 7, 279–285 (2000).zbMATHMathSciNetGoogle Scholar
  134. 134.
    J. Kedra and D. McDuff, Homotopy properties of Hamiltonian group actions, E-print math.SG/0404539 (2004), http://xxx.lanl.gov.Google Scholar
  135. 135.
    B. A. Khesin and Yu. V. Chekanov, “Invariants of the Euler equation for the ideal or barotropic hydrodynamics and superconductivity in D dimension,” Phys. D, 40, No. 1, 119–131 (1989).zbMATHMathSciNetGoogle Scholar
  136. 136.
    A. A. Kirillov, Elements of Representation Theory [in Russian], Nauka, Moscow (1972).Google Scholar
  137. 137.
    A. A. Kirillov, “Infinite-dimensional Lie groups: Their orbits, invariants and representations. The geometry of moments,” Lect. Notes Math., 970, 101–123 (1982).MathSciNetGoogle Scholar
  138. 138.
    A. A. Kirillov, “Kähler structure on K-orbits of diffeomorphisms group of the circle,” Funkts. Anal. Prilozh., 21, No. 2, 42–45 (1987).MathSciNetGoogle Scholar
  139. 139.
    A. A. Kirillov, “The orbit method. I: Geometric quantization; II: Infinite-dimensional Lie groups and Lie algebras,” Contemp. Math., 145, 1–63 (1993).MathSciNetGoogle Scholar
  140. 140.
    A. A. Kirillov and D. V. Yur’ev, “Kähler geometry of the infinite-dimensional homogeneous space M = Diff+(S 1)/S 1,” Funkts. Anal. Prilozh., 21, No. 4, 35–46 (1987).MathSciNetGoogle Scholar
  141. 141.
    S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vols. 1, 2 [Russian translation], Nauka, Moscow (1981).Google Scholar
  142. 142.
    O. Kobayashi, A. Yoshioka, Y. Maeda, and H. Omori, “The theory of infinite-dimensional Lie groups and its applications,” Acta Appl. Math., 3, No. 1, 71–106 (1985).zbMATHMathSciNetGoogle Scholar
  143. 143.
    N. Kopell, “Commuting diffeomorphisms,” Proc. Symp. Pure Math., 14, Amer. Math. Soc., Providence, Rhode Island (1970), pp. 165–184.Google Scholar
  144. 144.
    B. Kostant, “Quantization and unitary representations,” Lect. Notes Math., 170, 87–208 (1970).MathSciNetGoogle Scholar
  145. 145.
    F. Lalonde and D. McDu., “The geometry of symplectic energy,” Ann. Math., 141, No. 2, 319–333 (1995).Google Scholar
  146. 146.
    F. Lalonde, D. McDuff, and L. Polterovich, “On the flux conjectures,” CRM Proc. Lect. Notes, 15, Amer. Math. Soc., Providence, Rhode Island (1998), pp. 69–85.Google Scholar
  147. 147.
    F. Lalonde, D. McDuff, and L. Polterovich, “Topological rigidity of Hamiltonian loops and quantum cohomology,” Invent. Math., 135, 369–385 (1999).MathSciNetGoogle Scholar
  148. 148.
    P. F. Lam, “Embedding a homeomorphism in a flow subject to differentiability conditions,” in: Topological Dynamics, Benjamin, New York (1968), pp. 319–333.Google Scholar
  149. 149.
    L. D. Landau and E. M. Lifshits, Theoretical Physics, Vol. 3, Quantum Mechanics [in Russian], Nauka, Moscow (1989).Google Scholar
  150. 150.
    S. Lang, Introduction to Differentiable Manifolds, New York (1962).Google Scholar
  151. 151.
    J. Leslie, “On a differential structure for the group of diffeomorphisms,” Topology, 6, 263–271 (1967).zbMATHMathSciNetGoogle Scholar
  152. 152.
    M. V. Losik, “On Frech’et spaces as diffeologic spaces,” Izv. Vyssh. Ucheb. Zaved., Ser. Mat., 5, 36–42 (1992).MathSciNetGoogle Scholar
  153. 153.
    A. M. Lukatskii, “Algebras of vector fields and diffeomorphism groups of compact manifolds,” Funkts. Anal. Prilozh., 8, No. 2, 87–88 (1974).Google Scholar
  154. 154.
    A. M. Lukatskii, “On generator systems in diffeomorphism groups of compact manifolds,” Dokl. Akad. Nauk SSSR, 220, No. 2, 285–288 (1975).MathSciNetGoogle Scholar
  155. 155.
    A. M. Lukatskii, “On homogeneous vector bundles and diffeomorphism groups of compact homogeneous spaces, Izv. Akad. Nauk SSSR, Ser. Mat., 39, 1274–1285 (1975).MathSciNetGoogle Scholar
  156. 156.
    A. M. Lukatskii, “On the structure of the Lie algebra of spherical vector fields and diffeomorphism groups,” Sib. Mat. Zh., 18, No. 1, 161–173 (1977).Google Scholar
  157. 157.
    A. M. Lukatskii, “Finite generation of diffeomorphism groups,” Usp. Mat. Nauk, 23, No. 1, 219–220 (1978).Google Scholar
  158. 158.
    A. M. Lukatskii, “On generator systems in the diffeomorphism group of the n-dimensional torus,” Mat. Zametki, 26, No. 1, 27–34 (1979).MathSciNetGoogle Scholar
  159. 159.
    A. M. Lukatsky, “Construction of finite systems of generators for the Lie algebras of vector fields for group of diffeomorphisms of compact manifolds,” Select. Math. Sov., 1, No. 2, 185–195 (1981).Google Scholar
  160. 160.
    A. M. Lukatskii, “On the curvature of the measure-preserving diffeomorphism group of the two-dimensional sphere,” Funkts. Anal. Prilozh., 13, No. 3, 23–27 (1979).MathSciNetGoogle Scholar
  161. 161.
    A. M. Lukatskii, “On the curvature of the measure-preserving diffeomorphism group of the ndimensional torus,” Sib. Math. Zh., 25, No. 6, 76–88 (1984).MathSciNetGoogle Scholar
  162. 162.
    A. M. Lukatskii, “On the structure of the curvature tensor of the measure-preserving diffeomorphism group of a compact two-dimensional manifold,” Sib. Mat. Zh., 29, No. 6, 95–99 (1988).MathSciNetGoogle Scholar
  163. 163.
    A. M. Lukatsky, “On the curvature of the diffeomorphisms group,” Ann. Global Anal. Geom., 11, 135–140 (1993).zbMATHMathSciNetGoogle Scholar
  164. 164.
    J. N. Mather, “Commutators of diffeomorphisms,” Comment. Math. Helv., 49, 512–528 (1974).zbMATHMathSciNetGoogle Scholar
  165. 165.
    J. N. Mather, “Commutators of diffeomorphisms, II,” Comment. Math. Helv., 50, 33–40 (1975).zbMATHMathSciNetGoogle Scholar
  166. 166.
    J. N. Mather, “A curious remark concerning the geometric transfer map,” Comment. Math. Helv., 59, 86–110 (1984).zbMATHMathSciNetGoogle Scholar
  167. 167.
    J. N. Mather, “Commutators of diffeomorphisms, III,” Comment. Math. Helv., 60, No. 1, 122–124 (1985).zbMATHMathSciNetGoogle Scholar
  168. 168.
    D. McDuff, “The lattice of normal subgroups of the group of diffeomorphisms or homeomorphisms of an open manifold,” J. London Math. Soc., 18, No. 2, 353–364 (1978).MathSciNetGoogle Scholar
  169. 169.
    D. McDuff, “The homology of some groups of diffeomorphisms,” Comment. Math. Helv., 55, 97–120 (1980).zbMATHMathSciNetGoogle Scholar
  170. 170.
    D. McDuff, “Local homology of groups of volume-preserving diffeomorphisms, I,” Ann. Sci. Éc. Norm. Super., 15, 609–648 (1982).zbMATHMathSciNetGoogle Scholar
  171. 171.
    D. McDuff, “Local homology of groups of volume-preserving diffeomorphisms, II,” Comment. Math. Helv., 58, 135–165 (1983).zbMATHMathSciNetGoogle Scholar
  172. 172.
    D. McDuff, “Local homology of groups of volume-preserving diffeomorphisms, III,” Ann. Sci. Éc. Norm. Super., Ser. 4, 16, 529–540 (1983).zbMATHMathSciNetGoogle Scholar
  173. 173.
    D. McDuff, “Some canonical cohomology classes on groups of volume preserving diffeomorphisms,” Trans. Amer. Math. Soc., 275, No. 1, 345–356 (1983).zbMATHMathSciNetGoogle Scholar
  174. 174.
    D. McDuff, “Symplectic diffeomorphisms and the flux homomorphism,” Invent. Math., 77, No. 2, 353–366 (1984).zbMATHMathSciNetGoogle Scholar
  175. 175.
    D. McDuff, “Remarks on the homotopy type of groups of symplectic diffeomorphisms,” Proc. Amer. Math. Soc., 94, No. 2, 348–352 (1985).zbMATHMathSciNetGoogle Scholar
  176. 176.
    D. McDuff, “The moment map for circle actions on symplectic manifolds,” J. Geom. Phys., 5, 149–161 (1988).zbMATHMathSciNetGoogle Scholar
  177. 177.
    D. McDuff, Lectures on groups of symplectomorphisms, E-print mathDG/0201032 (2002), http://xxx.lanl.gov.Google Scholar
  178. 178.
    D. McDuff, A survey of the topological properties of symplectomorphism groups, E-print math.SG/0404340 (2004), http://xxx.lanl.gov.Google Scholar
  179. 179.
    J. Marsden, D. Ebin, and A. Fisher, “Diffeomorphism groups, hydrodynamics, and relativity,” in: 13th Biennial Seminar of Canad. Math. Congress (J. Vanstone, ed.), Montreal (1972), pp. 135–279.Google Scholar
  180. 180.
    J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, Springer-Verlag (1999).Google Scholar
  181. 181.
    J. E. Marsden, T. S. Ratiu, and S. Shkoller, “The geometry and analysis of the averaged Euler equations and a new diffeomorphism group,” Geom. Funct. Anal., 10, 582–599 (2000).zbMATHMathSciNetGoogle Scholar
  182. 182.
    J. Marsden, T. Ratiu, and A. Weinstein, “Semidirect products and reduction in mechanics,” Trans. Amer. Math. Soc., 281, No. 1, 147–177 (1984).zbMATHMathSciNetGoogle Scholar
  183. 183.
    J. Marsden and A. Weinstein, “The Hamiltonian structure of the Maxwell-Vlasov equations,” Phys. D, 4, 394–406 (1982).MathSciNetGoogle Scholar
  184. 184.
    J. E. Marsden and A. Weinstein, “Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids,” Phys. D, 7, 305–323 (1983).MathSciNetGoogle Scholar
  185. 185.
    J. N. Mather, “Simplicity of certain groups of diffeomorphisms,” Bull. Amer. Math. Soc., 80, No. 2, 211–273 (1974).MathSciNetGoogle Scholar
  186. 186.
    W. Michor, “The cohomology of the diffeomorphism group of a manifold is a Gelfand-Fuks cohomology,” Rend. Circ. Mat. Palermo, 36, Suppl. 14, 235–246 (1987).MathSciNetGoogle Scholar
  187. 187.
    W. Michor and C. Vizman, “n-Transitivity of certain diffeomorphism groups,” Acta Math. Univ. Comenianae, 63, No. 2, 1–4 (1994).MathSciNetGoogle Scholar
  188. 188.
    J. W. Milnor, “On spaces having the homotopy type of a CW complex,” Trans. Amer. Math. Soc., 90, 272–280 (1959).MathSciNetGoogle Scholar
  189. 189.
    J. W. Milnor, “Remarks on infinite-dimensional Lie groups,” in: Relativity, Groups, and Topology, II (B. S. de Witt and R. Stora, eds.), North-Holland, Amsterdam (1984), pp. 1007–1058.Google Scholar
  190. 190.
    A. S. Mishchenko and A. T. Fomenko, “Euler equations on finite-dimensional Lie groups,” Izv. Akad. Nauk SSSR, Ser. Mat., 42, No. 2, 396–415 (1978).zbMATHMathSciNetGoogle Scholar
  191. 191.
    G. Misiolek, “Stability of flows of ideal fluids and the geometry of the group of diffeomorphisms,” Indiana Univ. Math. J., 2, 215–235 (1993).MathSciNetGoogle Scholar
  192. 192.
    G. Misiolek, “Conjugate points in \(\mathcal{D}_\mu (T^2 )\),” Proc. Amer. Math. Soc., 124, 977–982 (1996).zbMATHMathSciNetGoogle Scholar
  193. 193.
    G. Misiolek, “A shallow water equation as a geodesic flow on the Bott-Virasoro group,” J. Geom. Phys., 24, 203–208 (1998).zbMATHMathSciNetGoogle Scholar
  194. 194.
    G. Misiolek, “The exponential map on the free loop spaces is Fredholm,” Geom. Funct. Anal., 7, 954–969 (1997).zbMATHMathSciNetGoogle Scholar
  195. 195.
    D. Montgomery and L. Zippin, Transformation Groups, Interscience, New York (1955).zbMATHGoogle Scholar
  196. 196.
    T. Morimoto and N. Tanaka, “The classification of real primitive infinite Lie algebras,” J. Math. Kyoto Univ., 10, 207–243 (1970).zbMATHMathSciNetGoogle Scholar
  197. 197.
    J. Moser, “On the volume elements on a manifold,” Trans. Amer. Math. Soc., 120, 286–294 (1965).zbMATHMathSciNetGoogle Scholar
  198. 198.
    S. Nag and A. Verjovsky, “Diff(S 1) and the Teichmuller spaces,” Commun. Math. Phys., 130, No. 1, 123–138 (1990).zbMATHMathSciNetGoogle Scholar
  199. 199.
    F. Nakamura, Y. Hattori, and T. Kambe, “Geodesics and curvature of a group of diffeomorphisms and motion of an ideal fluid,” J. Phys. A: Math. Gen., 25, L45–L50 (1992).MathSciNetGoogle Scholar
  200. 200.
    N. Nakanishi, “On the structure of infinite transitive primitive Lie algebras,” Proc. Jpn. Acad., 52, 14–16 (1976).zbMATHMathSciNetGoogle Scholar
  201. 201.
    R. Narasimhan, Analysis on Real and Complex Manifolds [Russian translation], Mir, Moscow (1971).Google Scholar
  202. 202.
    Z. Nitecki, Differentiable Dynamics. An Introduction to the Orbit Structure of Diffeomorphisms, The MIT Press (1971).Google Scholar
  203. 203.
    H. Omori, “On the group of diffeomorphisms on a compact manifold,” Proc. Symp. Pure Math., 15, 167–183 (1970).MathSciNetGoogle Scholar
  204. 204.
    H. Omori, “Local structures of groups of diffeomorphisms,” J. Math. Soc. Jpn., 24, No. 1, 60–88 (1972).zbMATHMathSciNetGoogle Scholar
  205. 205.
    H. Omori, “On smooth extension theorems,” J. Math. Soc. Jpn., 24, No. 3, 405–432 (1972).zbMATHMathSciNetGoogle Scholar
  206. 206.
    H. Omori, “Group of diffeomorphisms and their subgroups,” Trans. Amer. Math. Soc., 179, 85–122 (1973).zbMATHMathSciNetGoogle Scholar
  207. 207.
    H. Omori, Infinite-Dimensional Lie Transformations Groups, Lect. Notes Math., 427 (1974).Google Scholar
  208. 208.
    H. Omori and P. Harpe, “About interactions between Banach-Lie groups and finite-dimensional manifolds,” J. Math. Kyoto Univ., 12, No. 3, 543–570 (1972).zbMATHMathSciNetGoogle Scholar
  209. 209.
    K. Ono, “Some remarks on group actions in symplectic geometry,” J. Fac. Sci. Univ. Tokyo, Sec. IA, 35, 431–437 (1988).zbMATHGoogle Scholar
  210. 210.
    K. Ono, “Equivariant projective imbeddings theorem for symplectic manifolds,” J. Fac. Sci. Univ. Tokyo, Sec. IA, 35, 381–392 (1988).zbMATHGoogle Scholar
  211. 211.
    V. Yu. Ovsienko, B. A. Khesin, and Yu. V. Chekanov, “Integrals of the Euler equations in multidimensional hydrodynamics and superconductivity,” J. Sov. Math., 59, No. 5, 1096–1102 (1992).MathSciNetGoogle Scholar
  212. 212.
    R. Palais, “Homotopy theory of infinite-dimensional manifolds,” Topology, 5, 1–16 (1966).zbMATHMathSciNetGoogle Scholar
  213. 213.
    R. Palais, Foundations of Global Nonlinear Analysis, Benjamin, New York (1968).Google Scholar
  214. 214.
    R. Palais, Seminar on the Atiyah-Singer Index Theorem [Russian translation], Mir, Moscow (1970).zbMATHGoogle Scholar
  215. 215.
    R. Palais and T. E. Stewart, “The cohomology of differentiable transformation groups,” Amer. J. Math., 83, No. 4, 623–644 (1961).zbMATHMathSciNetGoogle Scholar
  216. 216.
    J. Palis, “Vector fields generate few diffeomorphisms,” Bull. Amer. Math. Soc., 80, No. 3, 503–505 (1974).zbMATHMathSciNetGoogle Scholar
  217. 217.
    J. Palis and J. C. Yoccoz, “Rigidity of centralizers of diffeomorphisms,” Ann. Sci. Éc. Norm. Super., Ser. 4, 22, 81–98 (1989).zbMATHMathSciNetGoogle Scholar
  218. 218.
    M. A. Parinov, “On the groups of diffeomorphism preserving nondegenerate analytic covector fields,” Mat. Sb., 186, No. 5, 115–126 (1995).MathSciNetGoogle Scholar
  219. 219.
    J. F. Plante, “Diffeomorphisms without periodic points,” Proc. Amer. Math. Soc., 88, 716–718 (1983).zbMATHMathSciNetGoogle Scholar
  220. 220.
    A. Pressly and G. Segal, Loop Groups, Oxford Math. Monogr., Clarendon Press, Oxford (1988).Google Scholar
  221. 221.
    T. Ratiu and R. Schmid, “The differentiable structure of three remarkable diffeomorphisms groups,” Math. Z., 177, 81–100 (1981).zbMATHMathSciNetGoogle Scholar
  222. 222.
    A. Reznikov, “Continuous cohomology of the group of volume-preserving and symplectic diffeomorphisms, measurable transfer and higher asymptotic cycles,” Select. Math. New Ser., 5, 181–198 (1999).zbMATHMathSciNetGoogle Scholar
  223. 223.
    P. Rouchon, “The Jacobi equation, Riemannian curvature, and the motion of a perfect incompressible fluid,” Eur. J. Mech., 11, No. 3, 317–336 (1992).zbMATHMathSciNetGoogle Scholar
  224. 224.
    W. Rudin, Mathematical Analysis [Russian translation], Mir, Moscow (1975).Google Scholar
  225. 225.
    T. Rybicki, “A note on groups of symplectomorphisms,” Ann. Sci. Math. Pol., Ser. I, 38, 115–126 (1998).zbMATHMathSciNetGoogle Scholar
  226. 226.
    E. Shavgulidze “Quasi-invariant measures on diffeomorphism groups,” Tr. Mat. Inst. Ross. Akad. Nauk, 217, 189–208 (1997).MathSciNetGoogle Scholar
  227. 227.
    E. V. Shchepin, “Hausdorff dimension and dynamics of diffeomorphisms,” Mat. Zametki, 65, No. 3, 457–463 (1999).MathSciNetGoogle Scholar
  228. 228.
    L. I. Sedov, Continuous-Medium Mechanics, Vol. 1 [in Russian], Mir, Moscow (1973).Google Scholar
  229. 229.
    A. G. Sergeev, Kahler Geometry of Loop Spaces [in Russian], Moscow (2001).Google Scholar
  230. 230.
    D. Serre, “Invariants et degenerescence symplectique de l’equation d’Euler des fluids parfaits incompressibles,” C. R. Acad. Sci. Paris, Ser. A, 298, 349 (1984).zbMATHMathSciNetGoogle Scholar
  231. 231.
    H. Shimomura, “Quasi-invariant measures on the group of diffeomorphisms and smooth vectors of unitary representations,” J. Funct. Anal., 187, 406–441 (2001).zbMATHMathSciNetGoogle Scholar
  232. 232.
    S. Shkoller, “Geometry and curvature of diffeomorphism groups with H 1 metric and mean hydrodynamics,” J. Funct. Anal., 160, 337–365 (1998).zbMATHMathSciNetGoogle Scholar
  233. 233.
    S. Shkoller, Groups of diffeomorphisms for manifolds with boundary and hydrodynamics, Preprint (1999).Google Scholar
  234. 234.
    S. Shnider, “The classification of real primitive infinite Lie algebras,” J. Differ. Geom., 4, 81–89 (1970).zbMATHMathSciNetGoogle Scholar
  235. 235.
    A. I. Shnirelman, “On geometry of the diffeomorphism group and dynamics of ideal incompressible fluid,” Mat. Sb., 128, No. 1, 82–109 (1985).MathSciNetGoogle Scholar
  236. 236.
    A. Shnirelman, “Attainable diffeomorphisms,” Geom. Funct. Anal., 3, No. 3, 297–294 (1993).MathSciNetGoogle Scholar
  237. 237.
    A. Shnirelman, “Generalized fluid flows, their approximation and applications,” Geom. Funct. Anal., 4, No. 5, 586–620 (1994).zbMATHMathSciNetGoogle Scholar
  238. 238.
    A. Shnirelman, “Evolution of singularities, generalized Liapunov function and generalized integral for an ideal incompressible fluid,” Amer. J. Math., 119, No. 3, 579–608 (1997).zbMATHMathSciNetGoogle Scholar
  239. 239.
    I. M. Singer and S. Sternberg, “On the infinite groups of Lie and Cartan, I,” J. Anal. Math., 15, 1–114 (1965).zbMATHMathSciNetGoogle Scholar
  240. 240.
    S. Smale, “Diffeomorphisms of the 2-sphere,” Proc. Amer. Math. Soc., 10, 621–626 (1959).zbMATHMathSciNetGoogle Scholar
  241. 241.
    S. Smale, “A survey of some recent developements in differential topology,” Bull. Amer. Math. Soc., 69, 131–185 (1963).MathSciNetGoogle Scholar
  242. 242.
    S. Smale, “Differentiable dynamics systems,” Bull. Amer. Math. Soc., 73, 747–817 (1967).MathSciNetGoogle Scholar
  243. 243.
    S. Smale, “Topology and mechanics,” Usp. Mat. Nauk, 27, No. 2, 77–133 (1972).MathSciNetGoogle Scholar
  244. 244.
    N. K. Smolentsev, “First integrals of ideal barotropic fluid flows,” in: All-Russian Conference on Contemporary Problems in Geometry, Abstracts of Reports [in Russian], Minsk (1979), p. 182.Google Scholar
  245. 245.
    N. K. Smolentsev, “On the Maupertuis principle,” Sib. Mat. Zh., 20, No. 5, 1092–1098 (1979).MathSciNetGoogle Scholar
  246. 246.
    N. K. Smolentsev, “On a certain weak Riemannian structure on the diffeomorphism group,” Izv. Vyssh. Ucheb. Zaved., Ser. Mat., 5, 78–80 (1979).MathSciNetGoogle Scholar
  247. 247.
    N. K. Smolentsev, “Integrals of ideal barotropic fluid flows,” Sib. Math. Zh., 23, No. 1, 205–208 (1982).zbMATHMathSciNetGoogle Scholar
  248. 248.
    N. K. Smolentsev, “Bi-invariant metric on the diffeomorphism group of a three-dimensional manifold,” Sib. Mat. Zh., 24, No. 1, 152–159 (1983).MathSciNetGoogle Scholar
  249. 249.
    N. K. Smolentsev, “On the group of diffeomorphisms leaving a vector field fixed,” Sib. Mat. Zh., 25, No. 2, 180–185 (1984).MathSciNetGoogle Scholar
  250. 250.
    N. K. Smolentsev, “On the vector product on a seven-dimensional manifold,” Sib. Math. Zh., 25, No. 5, 157–167 (1984).MathSciNetGoogle Scholar
  251. 251.
    N. K. Smolentsev, “Bi-invariant metrics on certain diffeomorphism groups,” in: Function Theory and Its Applications, Collection of Scientific Works [in Russian], Kemerovo (1985), pp. 73–78.Google Scholar
  252. 252.
    N. K. Smolentsev, “Bi-invariant metrics on the symplectic diffeomorphism group and the equation \(\frac{\partial }{{\partial t}}\Delta F = \{ \Delta F,F\} \),” Sib. Mat. Zh., 27, No. 1, 150–156 (1986).MathSciNetGoogle Scholar
  253. 253.
    N. K. Smolentsev, “Geometric properties of the action of the exact symplectic diffeomorphism group on the space of associated metrics,” in: Geometry and Analysis [in Russian], Kemerovj (1991), pp. 31–36.Google Scholar
  254. 254.
    N. K. Smolentsev, “Curvature of the diffeomorphism group and volume element space,” Sib. Mat. Zh., 33, No. 4, 115–141 (1992).MathSciNetGoogle Scholar
  255. 255.
    N. K. Smolentsev, “Curvature of the classical diffeomorphism groups,” Sib. Mat. Zh., 74, No. 1, 169–176 (1994).MathSciNetGoogle Scholar
  256. 256.
    S. E. Stepanov and I. G. Shandra, “Seven classes of harmonic diffeomorphisms,” Mat. Zametki, 74, No. 5, 752–761 (2003).MathSciNetGoogle Scholar
  257. 257.
    S. E. Stepanov and I. G. Shandra, “Geometry of infinitesimal harmonic transformations,” Ann. Global Anal. Geom., 24, No. 3, 291–299 (2003).zbMATHMathSciNetGoogle Scholar
  258. 258.
    S. Sternberg, Lectures on Differential Geometry, Prentice Hall, Englewood Cliffs, New Jersey (1964).zbMATHGoogle Scholar
  259. 259.
    F. Takens, “Characterization of a differentiable structure by its group of diffeomorphisms,” Bol. Soc. Bras. Math., 10, No. 1, 17–26 (1979).zbMATHMathSciNetGoogle Scholar
  260. 260.
    W. Thurston, “Foliations and groups of diffeomorphisms,” Bull. Amer. Math. Soc., 80, No. 2, 04–307 (1974).MathSciNetGoogle Scholar
  261. 261.
    A. M. Vershik, I. M. Gel’fand, and M. I. Graev, “Representations of diffeomorphism groups,” Usp. Mat. Nauk, 30, No. 6, 3–50 (1975).zbMATHGoogle Scholar
  262. 262.
    A. M. Vershik, “Description of invariant measures for actions of certain infinite-dimensional groups,” Dokl. Akad. Nauk SSSR, 218, No. 4, 749–752 (1974).MathSciNetGoogle Scholar
  263. 263.
    N. Ya. Vilenkin, Special Functions and Group Representation Theory [in Russian], Nauka, Moscow (1965).Google Scholar
  264. 264.
    A. M. Vinogradov and I. S. Krasil’shchik, “What is Hamiltonian formalism?” Usp. Mat. Nauk, 30, No. 1, 173–198 (1975).Google Scholar
  265. 265.
    A. M. Vinogradov and B. A. Kupershmidt, “Structure of Hamiltonian mechanics,” 32, No. 4, 175–236 (1977).zbMATHMathSciNetGoogle Scholar
  266. 266.
    C. Vizman, Coadjoint orbits in infinite dimensions, Preprint (1995).Google Scholar
  267. 267.
    N. Watanabe, “Existence of volume preserving diffeomorphisms without periodic points on three-dimensional manifolds,” Proc. Amer. Math. Soc., 97, No. 4, 724–726 (1986).zbMATHMathSciNetGoogle Scholar
  268. 268.
    A. Weinstein, Lectures on Symplectic Manifolds, Amer. Math. Soc. Conf. Board., Reg. Conf. Math., 29, Providence, Rhode Island (1977).Google Scholar
  269. 269.
    M. Wolf, “The Teichmuller theory of harmonic maps,” J. Differ. Geom., 29, No. 2, 449–479 (1989).zbMATHGoogle Scholar
  270. 270.
    T. Yagasaki, Homotopy types of diffeomorphism groups of noncompact 2-manifolds, E-print math.GT/0109183 (2001), http://xxx.lanl.gov.Google Scholar
  271. 271.
    S. Yamada, “Weil-Peterson convexity of the energy functional on classical and universal Teichmuller spaces,” J. Differ. Geom., 51, 35–96 (1999).zbMATHGoogle Scholar
  272. 272.
    K. Yoshida, “Riemannian curvature on the group of area-preserving diffeomorphisms (motions of fluid) on 2-sphere,” Phys. D, 100, Nos. 3–4, 377–389 (1997).zbMATHMathSciNetGoogle Scholar
  273. 273.
    V. A. Zaitseva, V. V. Kruglov, A. G. Sergeev, M. S. Strigunova, and K. A. Trushkin, “Remarks on loop groups of compact Lie groups and the diffeomorphism group of the circle,” Tr. Mat. Inst. Ross. Akad. Nauk, 224 (1999).Google Scholar
  274. 274.
    V. Zeitlin and T. Kambe, “Two-dimensional ideal magnetohydrodynamics and differential geometry,” J. Phys. A: Math. Gen., 26, 5025–5031 (1993).zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  • N. K. Smolentsev

There are no affiliations available

Personalised recommendations