Diffeomorphism groups of compact manifolds

Journal of Mathematical Sciences

Volumn 146, Issue 6, 2007, Pages 6213-6312

  Smolentsev, N K ^a  

^a NONE

Author keywords

[No Author keywords available]

Indexed keywords

EID: 35148841975     PISSN: 10723374     EISSN: 15738795     Source Type: Journal    
DOI: 10.1007/s10958-007-0471-0     Document Type: Article

References (274)

1. K. Abe and K. Fukui, "On commutators of equivariant diffeomorphisms," Proc. Jpn. Acad., 54, 52-54 (1978).
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.
3. K. Abe, "On the homotopy type of groups of equivariant diffeomorphisms," Publ. RIMS Kyoto Univ., 16, 601-626 (1980).
4. R. Abraham, Lectures of Smale on Differential Topology, Mimeographed notes, Columbia Univ., New York (1962).
5. R. Abraham and J. Marsden, Foundations of Mechanics, Benjamin, New York (1967).
6. M. Adams, T. Ratiu, and R. Schmid, "A Lie group structure for pseudodifferential operators," Math. Ann., 273, No. 4, 529-551 (1986).
7. M. Adams, T. Ratiu, and R. Schmid, "A Lie group structure for Fourier integral operators," Math. Ann., 276, No. 1, 19-41 (1986).
8. G. D'Ambra and M. Gromov, "Lecture on transformation groups: Geometry and dynamics," Surv. Differ. Geom., 1, 19-111 (1991).
9. P. Antonelli, D. Burghelea, and P. Kahn, "The non-infinite homotopy type of some diffeomorphism groups," Topology, 11, 1-49 (1972).
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).
11. V. I. Arnold, "Variational principle for three-dimensional stationary flows of the ideal fluid," Prikl. Mat. Mekh., 29, No. 5, 846-851 (1965).
12. V. I. Arnold, "Sur la topologie des écoulements stationnaires des fluides parfaits," C. R. Acad. Sci. Paris, 261, 117-120 (1965).
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).
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).
15. V. I. Arnold, Mathematical Methods of Classical Mechanics [in Russian], Nauka, Moscow (1974).
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.
17. V. I. Arnold, "The asymptotic Hopf invariant and its applications," Select. Math. Sov., 5, 327-345 (1986).
18. V. I. Arnold, "The asymptotic Hopf invariant and its applications," Select. Math. Sov., 5, No. 4, 327-345 (1986).
19. V. I. Arnold and B. Khesin Topological Methods in Hydrodynamics, Springer Verlag, New York (1998).
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).
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).
22. V. I. Averbukh and O. G. Smolyanov, "Differentiation theory in linear topological spaces," Usp. Math. Nauk, 22, No. 6, 201-260 (1967).
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).
24. A. Banyaga, "On the group of equivariant diffeomorphisms," Topology, 16, 279-283 (1977).
25. A. Banyaga, "Sur la structure du groupe des diffeomorphismes qui preservent une forme symplectique," Comment. Math. Helvet., 53, 174-227 (1978).
26. A. Banyaga, "The group of diffeomorphisms preserving a regular contact form," Monogr. Enseign. Math., 26, 47-53 (1978).
27. 2n+1," Bort. Soc. Matem. Mexicana, 23, No. 2, 43-47 (1978).
28. A. Banyaga, "On fixed points of symplectic maps," Invent Math., 56, 215-229 (1980).
29. A. Banyaga, "Sur la cohomologie du groupe des diffeomorphismes, " C. R. Acad.Sci. Paris, Ser. I, 294, 625-627 (1982).
30. A. Banyaga, "On isomorphic classical diffeomorphism groups, II," J. Differ. Geom., 28, No. 1, 23-35 (1988).
31. A. Banyaga, "Sur la groupe des diffeomorphismes symplectiques," Lect. Notes. Math., 484, 50-56 (1975).
32. A. Banyaga, The Structure of Classical Diffeomorphisms Groups, Kluwer Academic Publ., Amsterdam (1997).
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).
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).
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).
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.
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).
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.
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).
40. A. Besse, Four-Dimensional Riemannian Geometry [Russian translation], Mir, Moscow (1985).
41. A. Besse, Einstein Manifolds, Vols. 1, 2 [Russian translation], Mir, Moscow (1990).
42. M. Bialy and L. Polterovich, "Hamiltonian diffeomorphisms and Lagrangian distribution," Geom. Funct. Anal., 2, No. 2, 173-210 (1992).
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).
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).
45. D. E. Blair, Contact Manifolds in Riemannian Geometry, Lect. Notes Math., 509, Springer-Verlag (1976).
46. W. M. Boothby, "The transitivity of the automorphisms of certain geometric structures," Trans. Amer. Math. Soc., 137, 93-100 (1969).
47. R. Bott, "On the characteristic classes of group of diffeomorphisms," Monogr. Enseign. Math., 26, 63-74 (1978).
48. 1) generalizing the Schwarzian derivative," Int. Math. Res. Notices, 1, 25-39 (1998).
49. R. Brooks, "Volumes and characteristic classes of foliations," Topology, 18, 295-304 (1979).
50. U. Bunke, Higher analytic torsion and cohomology of diffeomorphism groups, E-print dg-ga/9712001 (1997), http://xxx.lanl.gov.
51. 2n-1)," E-print math.DG/9802100 (1998), http://xxx.lanl.gov.
52. n ) and connected problems," Colloq. Int. CNRS, 7, No. 210, 3-17 (1973).
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.
54. J. Cheeger and J. Simons, "Differential characters and geometric invariants," Lect. Notes Math., 1167, 50-80 (1985).
55. S. Chern and J. Simons, "Characteristic forms and geometric invariants," Ann. Math., 99, No. 1, 48-69 (1974).
56. P. R. Chernnoff, "Irreducible representations of infinite-dimensional transformation groups and Lie algebras," Bull. Amer. Math. Soc., 13, No. 1, 46-48 (1985).
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.
58. A. Constantin and B. Kolev, "On the geometric approach to the motion inertial mechanical systems," J. Phys. A, 35, R51-R79 (2002).
59. A. Constantin and B. Kolev, "Geodesic flow on the diffeomorphism group of the circle," Comment. Math. Helv., 78, 787-804 (2003).
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).
61. A. A. Dezin, "Invariant forms and some structure properties of the Euler equations of hydrodynamics," Z. Anal. Anwend., 2, 401-409 (1983).
62. S. K. Donaldson, "Moment maps and diffeomorphisms," Asian J. Math., 3, No. 1, 1-16 (1999).
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).
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).
65. C. J. Earle and J. Eells, "A fibre bundle description of Teichmuller theory," J. Differ. Geom., 3, 19-43 (1969).
66. D. Ebin, "The manifold of Riemannian metrics," Proc. Symp. Pure Math., 15, 11-40 (1970).
67. D. Ebin, "Integrability of perfect fluid motion," Commun. Pure Appl. Math., 36, No. 1, 37-54 (1983).
68. D. Ebin and J. Marsden, "Groups of diffeomorphisms and the motion of an incompressible fluid," Ann. Math., 92, No. 1, 102-163 (1970).
69. J. Eells, "On the geometry of function spaces," in: Symp. Topology Algebra, Mexico (1958), pp. 303-307.
70. J. Eells, "On submanifolds of certain function spaces," Proc. Natl. Acad. Sci., 45, No. 10, 1520-1522 (1959).
71. J. Eells, "Alexander-Pontryagin duality in function spaces," Proc. Symp. Pure Math., 3, 109-129 (1961).
72. J. Eells, "A setting for global analysis," Bull. Amer. Math. Soc., 72, 751-787 (1966).
73. J. Eichhorn, "The manifold structure of maps between open manifolds," Ann. Global Anal. Geom., 11, 253-300 (1993).
74. J. Eichhorn, "Gauge theory on open manifolds of bounded geometry," Int. J. Mod. Phys., 7, 3927-3977 (1993).
75. J. Eichhorn, "Spaces of Riemannian metrics on open manifolds," Results Math., 27, 256-283 (1995).
76. J. Eichhorn and R. Schmid, "Form preserving diffeomorphisms on open manifolds," Ann. Global Anal. Geom., 14, 147-176 (1996).
77. J. Eichhorn, "Diffeomorphism groups on noncompact manifolds," Zap. Nauch. Semin. POMI, 234, 41-64 (1996).
78. J. Eichhorn and J. Fricke, "The module structure theorem for Sobolev spaces on open manifolds," Math. Nachr., 184, 35-47 (1998).
79. J. Eichhorn, "Poincaré's theorem and Teichmuller theory for open manifolds," Asian J. Math., 2, No. 2, 355-404 (1998).
80. J. Eichhorn, "A classification approach for open manifolds," Zap. Nauchn. Semin. POMI, 267, 9-45 (2000).
81. Ya. Eliashberg and L. Polterovich, "Bi-invariant metrics on the group of Hamiltinian diffeomorphisms," Int. J. Math., 4, No. 5, 727-738 (1993).
82. Ya. Eliashberg and T. Ratiu, "The diameter of the symplectomorphism group is infinite," Invent. Math., 103, No. 2, 327-340 (1991).
83. H. Eliasson, "On the geometry of manifold of maps," J. Differ. Geom., 1, 169-194 (1967).
84. D. B. A. Epstein, "The simplicity of certain groups of homeomorphisms," Compos. Math., 22, 165-173 (1970).
85. ∞ diffeomorphisms," Comment. Math. Helv., 59, 111-122 (1984).
86. J. Etnyre and R. Ghrist, Contact topology and hydrodynamics, II: Solid tori, E-print math.SG/9907112 (1999), http://xxx.lanl.gov.
87. J. Etnyre and R. Ghrist, Contact topology and hydrodynamics, III: Knotted flowlines, E-print math-ph/9906021 (1999), http://xxx.lanl.gov.
88. J. Etnyre and R. Ghrist, An index for closed orbits in Beltrami fields, E-print math.DS/0101095 (2001), http://xxx.lanl.gov.
89. R. P. Filipkewicz, "Isomorphisms between diffeomorphism groups," Ergodic Theor. Dynam. Syst., 2, 159-171 (1982).
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).
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).
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.
93. D. B. Fuks, Cohomologies of Infinite-Dimensional Lie Algebras [in Russian], Nauka, Moscow (1984).
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.
95. ∞ IE2, J. Math. Kyoto Univ., 15, No. 1, 201-210 (1975).
96. n, 0)," J. Math. Kyoto Univ., 20, 475-487 (1980).
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).
98. V. L. Ginzburg, "Some remarks on symplectic actions of compact groups," Math. Z., 210, 625-640 (1992).
99. Yu. E. Gliklikh, Analysis on Riemannian Manifolds and Problems of Mathematical Physics [in Russian], VGU, Voronezh (1989).
100. K. Godbillon, Differential Geometry and Analytic Mechanics [Russian translation], Mir, Moscow (1973).
101. M. Golubitsky and V. W. Guillemin, Stable Mappings and Their Singularities, Grad. Texts Math., 14, Springer-Verlag (1973).
102. J. Grabowski, "Free subgroups of diffeomorphisms groups," Fundam. Math., 131, No. 2, 103-121 (1988).
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).
104. R. E. Greene and K. Shiohama, "Diffeomorphisms and volume-preserving embeddings of noncompact manifolds," Trans. Amer. Math. Soc., 255, 403-414 (1979).
105. M. Gromov, "Pseudoholomorphic curves in symplectic manifolds," Invent. Math., 82, No. 2, 307-347 (1985).
106. M. Gromov, Partial Differential Relations [Russian translation], Mir, Moscow (1990).
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.
108. D. Gromol, W. Klingenberg, and W. Meyer, Riemannian Geometry in the Large [Russian translation], Mir, Moscow (1971).
109. V. W. Guillemin, "Infinite-dimensional primitive Lie algebras," J. Differ. Geom., 4, 257-282 (1970).
110. S. Haller and T. Rybicki, On the perfectness of nontransitive groups of diffeomorphisms, E-print math.DG/9902095 (1999), http://xxx.lanl.gov.
111. R. S. Hamilton, "The inverse function theorem of Nash and Moser," Bull. Amer. Math. Soc., 7, No. 1, 65-222 (1982).
112. D. Hart, "On the smoothness of generators," Topology, 22, No. 3, 357-363 (1983).
113. Y. Hatakeyama, "Some notes on the groups of automorphisms of contact and symplectic structures," Tohoku Math. J., 18, 338-347 (1966).
114. 3) O(4)," Ann. Math., 117, 553-607 (1983).
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.
116. Y. Hattori, "Ideal magnetohydrodynamics and passive scalar motion as geodesics on semidirect product groups," J. Phys. A: Math. Gen., 27, L21-L25 (1994).
117. ∞, isotopes al'identite, du tore de dimension n," C. R. Acad. Sci. Paris, Ser. A, 273, 232-234 (1971).
118. M. R. Herman, "Sur la groupe des diffeomorphismes du tore," Ann. Inst. Fourier, 23, No. 2, 75-86 (1973).
119. M. R. Herman, "Sur la conjugaison differentiable des diffeomorphismes du cercle a des rotations," Publ. Math. IHES, 49, 5-234 (1979).
120. M. W. Hirsch, Differential Topology, Grad. Texts Math., 33, Springer-Verlag (1976).
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).
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).
123. H. Hofer, "Estimates for the energy of a symplectic map," Comment. Math. Helv., 68, No. 1, 48-72 (1993).
124. H. Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics, Birkhauser, Basel-Boston-Berlin (1994).
125. 2 and the curvature of some infinite dimensional manifolds," Phys. Lett. B, 215, No. 4, 706-710 (1988).
126. D. Husemoller, Fibre Bundles, McGraw-Hill (1966).
127. R. S. Ismagilov, "On unitary representations of diffeomorphism groups of the circle," Functs. Anal. Prilozh., 5, No. 3, 45-54 (1971).
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).
129. n, n 2," Funkts. Anal. Prilozh., 9, 71-72 (1975).
130. ∞ (X,G), G = XU(2)," Mat. Sb., 100, No. 1, 117-131 (1976).
131. R. S. Ismagilov, "Unitary representations of the measure-preserving diffeomorphism group," Funkts. Anal. Prilozh., 1, No. 3, 80-81 (1977).
132. R. S. Ismagilov, "Inductive limits of the area-preserving diffeomorphism groups," Funkts. Anal. Prilozh., 37, No. 3, 36-50 (2003).
133. J. Kedra, "Remarks on the flux groups," Math. Res. Lett., 7, 279-285 (2000).
134. J. Kedra and D. McDuff, Homotopy properties of Hamiltonian group actions, E-print math.SG/0404539 (2004), http://xxx.lanl.gov.
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).
136. A. A. Kirillov, Elements of Representation Theory [in Russian], Nauka, Moscow (1972).
137. A. A. Kirillov, "Infinite-dimensional Lie groups: Their orbits, invariants and representations. The geometry of moments," Lect. Notes Math., 970, 101-123 (1982).
138. A. A. Kirillov, "Kähler structure on K-orbits of diffeomorphisms group of the circle," Funkts. Anal. Prilozh., 21, No. 2, 42-45 (1987).
139. A. A. Kirillov, "The orbit method. I: Geometric quantization; II: Infinite-dimensional Lie groups and Lie algebras," Contemp. Math., 145, 1-63 (1993).
140. 1," Funkts. Anal. Prilozh., 21, No. 4, 35-46 (1987).
141. S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vols. 1, 2 [Russian translation], Nauka, Moscow (1981).
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).
143. N. Kopell, "Commuting diffeomorphisms," Proc. Symp. Pure Math., 14, Amer. Math. Soc., Providence, Rhode Island (1970), pp. 165-184.
144. B. Kostant, "Quantization and unitary representations," Lect. Notes Math., 170, 87-208 (1970).
145. F. Lalonde and D. McDu., "The geometry of symplectic energy," Ann. Math., 141, No. 2, 319-333 (1995).
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.
147. F. Lalonde, D. McDuff, and L. Polterovich, "Topological rigidity of Hamiltonian loops and quantum cohomology," Invent. Math., 135, 369-385 (1999).
148. P. F. Lam, "Embedding a homeomorphism in a flow subject to differentiability conditions," in: Topological Dynamics, Benjamin, New York (1968), pp. 319-333.
149. L. D. Landau and E. M. Lifshits, Theoretical Physics, Vol. 3, Quantum Mechanics [in Russian], Nauka, Moscow (1989).
150. S. Lang, Introduction to Differentiable Manifolds, New York (1962).
151. J. Leslie, "On a differential structure for the group of diffeomorphisms," Topology, 6, 263-271 (1967).
152. M. V. Losik, "On Frech'et spaces as diffeologic spaces," Izv. Vyssh. Ucheb. Zaved., Ser. Mat., 5, 36-42 (1992).
153. A. M. Lukatskii, "Algebras of vector fields and diffeomorphism groups of compact manifolds," Funkts. Anal. Prilozh., 8, No. 2, 87-88 (1974).
154. A. M. Lukatskii, "On generator systems in diffeomorphism groups of compact manifolds," Dokl. Akad. Nauk SSSR, 220, No. 2, 285-288 (1975).
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).
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).
157. A. M. Lukatskii, "Finite generation of diffeomorphism groups," Usp. Mat. Nauk, 23, No. 1, 219-220 (1978).
158. A. M. Lukatskii, "On generator systems in the diffeomorphism group of the n-dimensional torus," Mat. Zametki, 26, No. 1, 27-34 (1979).
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).
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).
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).
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).
163. A. M. Lukatsky, "On the curvature of the diffeomorphisms group," Ann. Global Anal. Geom., 11, 135-140 (1993).
164. J. N. Mather, "Commutators of diffeomorphisms," Comment. Math. Helv., 49, 512-528 (1974).
165. J. N. Mather, "Commutators of diffeomorphisms, II," Comment. Math. Helv., 50, 33-40 (1975).
166. J. N. Mather, "A curious remark concerning the geometric transfer map," Comment. Math. Helv., 59, 86-110 (1984).
167. J. N. Mather, "Commutators of diffeomorphisms, III," Comment. Math. Helv., 60, No. 1, 122-124 (1985).
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).
169. D. McDuff, "The homology of some groups of diffeomorphisms," Comment. Math. Helv., 55, 97-120 (1980).
170. D. McDuff, "Local homology of groups of volume-preserving diffeomorphisms, I," Ann. Sci. Éc. Norm. Super., 15, 609-648 (1982).
171. D. McDuff, "Local homology of groups of volume-preserving diffeomorphisms, II," Comment. Math. Helv., 58, 135-165 (1983).
172. D. McDuff, "Local homology of groups of volume-preserving diffeomorphisms, III," Ann. Sci. Éc. Norm. Super., Ser. 4, 16, 529-540 (1983).
173. D. McDuff, "Some canonical cohomology classes on groups of volume preserving diffeomorphisms," Trans. Amer. Math. Soc., 275, No. 1, 345-356 (1983).
174. D. McDuff, "Symplectic diffeomorphisms and the flux homomorphism," Invent. Math., 77, No. 2, 353-366 (1984).
175. D. McDuff, "Remarks on the homotopy type of groups of symplectic diffeomorphisms," Proc. Amer. Math. Soc., 94, No. 2, 348-352 (1985).
176. D. McDuff, "The moment map for circle actions on symplectic manifolds," J. Geom. Phys., 5, 149-161 (1988).
177. D. McDuff, Lectures on groups of symplectomorphisms, E-print mathDG/0201032 (2002), http://xxx.lanl.gov.
178. D. McDuff, A survey of the topological properties of symplectomorphism groups, E-print math.SG/0404340 (2004), http://xxx.lanl.gov.
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.
180. J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, Springer-Verlag (1999).
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).
182. J. Marsden, T. Ratiu, and A. Weinstein, "Semidirect products and reduction in mechanics," Trans. Amer. Math. Soc., 281, No. 1, 147-177 (1984).
183. J. Marsden and A. Weinstein, "The Hamiltonian structure of the Maxwell-Vlasov equations," Phys. D, 4, 394-406 (1982).
184. J. E. Marsden and A. Weinstein, "Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids," Phys. D, 7, 305-323 (1983).
185. J. N. Mather, "Simplicity of certain groups of diffeomorphisms, " Bull. Amer. Math. Soc., 80, No. 2, 211-273 (1974).
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).
187. W. Michor and C. Vizman, "n-Transitivity of certain diffeomorphism groups," Acta Math. Univ. Comenianae, 63, No. 2, 1-4 (1994).
188. J. W. Milnor, "On spaces having the homotopy type of a CW complex," Trans. Amer. Math. Soc., 90, 272-280 (1959).
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.
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).
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).
192. G. Misiolek, "Conjugate points in, Proc. Amer. Math. Soc., 124, 977-982 (1996).
193. G. Misiolek, "A shallow water equation as a geodesic flow on the Bott-Virasoro group," J. Geom. Phys., 24, 203-208 (1998).
194. G. Misiolek, "The exponential map on the free loop spaces is Fredholm," Geom. Funct. Anal., 7, 954-969 (1997).
195. D. Montgomery and L. Zippin, Transformation Groups, Interscience, New York (1955).
196. T. Morimoto and N. Tanaka, "The classification of real primitive infinite Lie algebras," J. Math. Kyoto Univ., 10, 207-243 (1970).
197. J. Moser, "On the volume elements on a manifold," Trans. Amer. Math. Soc., 120, 286-294 (1965).
198. 1) and the Teichmuller spaces," Commun. Math. Phys., 130, No. 1, 123-138 (1990).
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).
200. N. Nakanishi, "On the structure of infinite transitive primitive Lie algebras," Proc. Jpn. Acad., 52, 14-16 (1976).
201. R. Narasimhan, Analysis on Real and Complex Manifolds [Russian translation], Mir, Moscow (1971).
202. Z. Nitecki, Differentiable Dynamics. An Introduction to the Orbit Structure of Diffeomorphisms, The MIT Press (1971).
203. H. Omori, "On the group of diffeomorphisms on a compact manifold," Proc. Symp. Pure Math., 15, 167-183 (1970).
204. H. Omori, "Local structures of groups of diffeomorphisms," J. Math. Soc. Jpn., 24, No. 1, 60-88 (1972).
205. H. Omori, "On smooth extension theorems," J. Math. Soc. Jpn., 24, No. 3, 405-432 (1972).
206. H. Omori, "Group of diffeomorphisms and their subgroups," Trans. Amer. Math. Soc., 179, 85-122 (1973).
207. H. Omori, Infinite-Dimensional Lie Transformations Groups, Lect. Notes Math., 427 (1974).
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).
209. K. Ono, "Some remarks on group actions in symplectic geometry," J. Fac. Sci. Univ. Tokyo, Sec. IA, 35, 431-437 (1988).
210. K. Ono, "Equivariant projective imbeddings theorem for symplectic manifolds," J. Fac. Sci. Univ. Tokyo, Sec. IA, 35, 381-392 (1988).
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).
212. R. Palais, "Homotopy theory of infinite-dimensional manifolds," Topology, 5, 1-16 (1966).
213. R. Palais, Foundations of Global Nonlinear Analysis, Benjamin, New York (1968).
214. R. Palais, Seminar on the Atiyah-Singer Index Theorem [Russian translation], Mir, Moscow (1970).
215. R. Palais and T. E. Stewart, "The cohomology of differentiable transformation groups," Amer. J. Math., 83, No. 4, 623-644 (1961).
216. J. Palis, "Vector fields generate few diffeomorphisms," Bull. Amer. Math. Soc., 80, No. 3, 503-505 (1974).
217. J. Palis and J. C. Yoccoz, "Rigidity of centralizers of diffeomorphisms," Ann. Sci. Éc. Norm. Super., Ser. 4, 22, 81-98 (1989).
218. M. A. Parinov, "On the groups of diffeomorphism preserving nondegenerate analytic covector fields," Mat. Sb., 186, No. 5, 115-126 (1995).
219. J. F. Plante, "Diffeomorphisms without periodic points," Proc. Amer. Math. Soc., 88, 716-718 (1983).
220. A. Pressly and G. Segal, Loop Groups, Oxford Math. Monogr., Clarendon Press, Oxford (1988).
221. T. Ratiu and R. Schmid, "The differentiable structure of three remarkable diffeomorphisms groups," Math. Z., 177, 81-100 (1981).
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).
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).
224. W. Rudin, Mathematical Analysis [Russian translation], Mir, Moscow (1975).
225. T. Rybicki, "A note on groups of symplectomorphisms," Ann. Sci. Math. Pol., Ser. I, 38, 115-126 (1998).
226. E. Shavgulidze "Quasi-invariant measures on diffeomorphism groups," Tr. Mat. Inst. Ross. Akad. Nauk, 217, 189-208 (1997).
227. E. V. Shchepin, "Hausdorff dimension and dynamics of diffeomorphisms," Mat. Zametki, 65, No. 3, 457-463 (1999).
228. L. I. Sedov, Continuous-Medium Mechanics, Vol. 1 [in Russian], Mir, Moscow (1973).
229. A. G. Sergeev, Kahler Geometry of Loop Spaces [in Russian], Moscow (2001).
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).
231. H. Shimomura, "Quasi-invariant measures on the group of diffeomorphisms and smooth vectors of unitary representations," J. Funct. Anal., 187, 406-441 (2001).
232. 1 metric and mean hydrodynamics," J. Funct. Anal., 160, 337-365 (1998).
233. S. Shkoller, Groups of diffeomorphisms for manifolds with boundary and hydrodynamics, Preprint (1999).
234. S. Shnider, "The classification of real primitive infinite Lie algebras," J. Differ. Geom., 4, 81-89 (1970).
235. A. I. Shnirelman, "On geometry of the diffeomorphism group and dynamics of ideal incompressible fluid," Mat. Sb., 128, No. 1, 82-109 (1985).
236. A. Shnirelman, "Attainable diffeomorphisms," Geom. Funct. Anal., 3, No. 3, 297-294 (1993).
237. A. Shnirelman, "Generalized fluid flows, their approximation and applications," Geom. Funct. Anal., 4, No. 5, 586-620 (1994).
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).
239. I. M. Singer and S. Sternberg, "On the infinite groups of Lie and Cartan, I," J. Anal. Math., 15, 1-114 (1965).
240. S. Smale, "Diffeomorphisms of the 2-sphere," Proc. Amer. Math. Soc., 10, 621-626 (1959).
241. S. Smale, "A survey of some recent developements in differential topology," Bull. Amer. Math. Soc., 69, 131-185 (1963).
242. S. Smale, "Differentiable dynamics systems," Bull. Amer. Math. Soc., 73, 747-817 (1967).
243. S. Smale, "Topology and mechanics," Usp. Mat. Nauk, 27, No. 2, 77-133 (1972).
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.
245. N. K. Smolentsev, "On the Maupertuis principle," Sib. Mat. Zh., 20, No. 5, 1092-1098 (1979).
246. N. K. Smolentsev, "On a certain weak Riemannian structure on the diffeomorphism group," Izv. Vyssh. Ucheb. Zaved., Ser. Mat., 5, 78-80 (1979).
247. N. K. Smolentsev, "Integrals of ideal barotropic fluid flows," Sib. Math. Zh., 23, No. 1, 205-208 (1982).
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).
249. N. K. Smolentsev, "On the group of diffeomorphisms leaving a vector field fixed," Sib. Mat. Zh., 25, No. 2, 180-185 (1984).
250. N. K. Smolentsev, "On the vector product on a seven-dimensional manifold," Sib. Math. Zh., 25, No. 5, 157-167 (1984).
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.
252. N. K. Smolentsev, "Bi-invariant metrics on the symplectic diffeomorphism group and the equation, Sib. Mat. Zh., 27, No. 1, 150-156 (1986).
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.
254. N. K. Smolentsev, "Curvature of the diffeomorphism group and volume element space," Sib. Mat. Zh., 33, No. 4, 115-141 (1992).
255. N. K. Smolentsev, "Curvature of the classical diffeomorphism groups," Sib. Mat. Zh., 74, No. 1, 169-176 (1994).
256. S. E. Stepanov and I. G. Shandra, "Seven classes of harmonic diffeomorphisms," Mat. Zametki, 74, No. 5, 752-761 (2003).
257. S. E. Stepanov and I. G. Shandra, "Geometry of infinitesimal harmonic transformations," Ann. Global Anal. Geom., 24, No. 3, 291-299 (2003).
258. S. Sternberg, Lectures on Differential Geometry, Prentice Hall, Englewood Cliffs, New Jersey (1964).
259. F. Takens, "Characterization of a differentiable structure by its group of diffeomorphisms," Bol. Soc. Bras. Math., 10, No. 1, 17-26 (1979).
260. W. Thurston, "Foliations and groups of diffeomorphisms," Bull. Amer. Math. Soc., 80, No. 2, 04-307 (1974).
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).
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).
263. N. Ya. Vilenkin, Special Functions and Group Representation Theory [in Russian], Nauka, Moscow (1965).
264. A. M. Vinogradov and I. S. Krasil'shchik, "What is Hamiltonian formalism?" Usp. Mat. Nauk, 30, No. 1, 173-198 (1975).
265. A. M. Vinogradov and B. A. Kupershmidt, "Structure of Hamiltonian mechanics," 32, No. 4, 175-236 (1977).
266. C. Vizman, Coadjoint orbits in infinite dimensions, Preprint (1995).
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).
268. A. Weinstein, Lectures on Symplectic Manifolds, Amer. Math. Soc. Conf. Board., Reg. Conf. Math., 29, Providence, Rhode Island (1977).
269. M. Wolf, "The Teichmuller theory of harmonic maps," J. Differ. Geom., 29, No. 2, 449-479 (1989).
270. T. Yagasaki, Homotopy types of diffeomorphism groups of noncompact 2-manifolds, E-print math.GT/0109183 (2001), http://xxx.lanl.gov.
271. S. Yamada, "Weil-Peterson convexity of the energy functional on classical and universal Teichmuller spaces," J. Differ. Geom., 51, 35-96 (1999).
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).
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).
274. V. Zeitlin and T. Kambe, "Two-dimensional ideal magnetohydrodynamics and differential geometry," J. Phys. A: Math. Gen., 26, 5025-5031 (1993).