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Diffeomorphism groups of compact manifolds. (English. Russian original) Zbl 1147.58012
J. Math. Sci., New York 146, No. 6, 6213-6312 (2007); translation from Sovrem. Mat. Prilozh. 37, 3-100 (2006).
This is a survey on diffeomorphism groups of manifolds viewed as infinite-dimensional Lie-Frechét groups. The intensive studies on diffeomorphism groups began after V. I. Arnold [Ann. Inst. Fourier 16, No. 1, 319–361 (1966; Zbl 0148.45301)] had shown that motions of the ideal incompressible fluid are geodesics on the group of volume-element-preserving diffeomorphisms. Many researchers (Omori, Ebin, Marsden, Hamilton, McDuff, Lukatskii, Shnirelman, Milnor, to mention a few) contributed to these studies, where the curvature and metric properties of the diffeomorphism groups, and application of the diffeomorphism groups to ideal fluid hydrodynamics were considered.
This work is divided into 13 parts.
Let \(M\) be a smooth compact orientable manifold without boundary of dimension \(n\). The set \(C^ 1{\mathcal D}\) consisting of \(C^ 1\)-diffeomorphisms of \(M\) is open in \(C^ 1(M,M)\) and is a topological group. After introductory information on differential operators and Lie derivatives, the author presents the inverse limit Hilbert (ILH) Lie group of diffeomorphisms on ILH-Lie and Hamilton Fréchet manifolds. In Part 3, various subgroups of the diffeomorphism group are studied: subgroups of strict ILH-Lie groups, groups of volume-element-preserving diffeomorphisms, symplectic diffeomorphism groups, contact transformation groups, and groups of diffeomorphisms leaving a vector field fixed. Next, exponential mappings and weak Riemannian structures on the diffeomorphism groups are defined.
Parts 6 and 7 deal with volume-element-preserving diffeomorphism groups and the ideal barotropic fluids.
The internal energy, Hamiltonian property of the hydromechanics equations, first integrals of motion of the ideal barotropic fluid, hydromechanical interpretations of geodesics, and Euler-Poincaré reduction are discussed. Curvatures of various groups of diffeomorphisms preserving the Riemannian volume element are calculated in Part 8. Symplectic and exact contact diffeomorphism groups and their properties are studied in Parts 10 and 11. Diffeomorphisms preserving certain tensor fields, harmonic diffeomorphisms, and diffeomorphism without periodic points are presented in Part 12. The last part deals with metric and topological properties of diffeomorphism groups: the diameter of the diffeomorphism group, symplectic transformation groups, algebraic properties, homotopy type, and homologies and cohomologies of diffeomorphism groups.
The list of references contains 274 entries.

MSC:
58D17 Manifolds of metrics (especially Riemannian)
53D05 Symplectic manifolds, general
76A99 Foundations, constitutive equations, rheology, hydrodynamical models of non-fluid phenomena
58-02 Research exposition (monographs, survey articles) pertaining to global analysis
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References:
[1] K. Abe and K. Fukui, ”On commutators of equivariant diffeomorphisms,” Proc. Jpn. Acad., 54, 52–54 (1978). · Zbl 0398.57032 · doi:10.3792/pjaa.54.52
[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. · Zbl 0978.58003
[3] K. Abe, ”On the homotopy type of groups of equivariant diffeomorphisms,” Publ. RIMS Kyoto Univ., 16, 601–626 (1980). · Zbl 0453.58013 · doi:10.2977/prims/1195187218
[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). · Zbl 0587.58047 · doi:10.1007/BF01472130
[7] M. Adams, T. Ratiu, and R. Schmid, ”A Lie group structure for Fourier integral operators,” Math. Ann., 276, No. 1, 19–41 (1986). · Zbl 0619.58010 · doi:10.1007/BF01450921
[8] G. D’Ambra and M. Gromov, ”Lecture on transformation groups: Geometry and dynamics,” Surv. Differ. Geom., 1, 19–111 (1991). · Zbl 0752.57017
[9] P. Antonelli, D. Burghelea, and P. Kahn, ”The non-infinite homotopy type of some diffeomorphism groups,” Topology, 11, 1–49 (1972). · Zbl 0225.57013 · doi:10.1016/0040-9383(72)90021-3
[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). · doi:10.1016/0370-2693(89)90916-7
[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). · Zbl 0623.57016
[18] V. I. Arnold, ”The asymptotic Hopf invariant and its applications,” Select. Math. Sov., 5, No. 4, 327–345 (1986). · Zbl 0623.57016
[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). · Zbl 0297.58008 · doi:10.1017/S0305004100049410
[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). · Zbl 0314.58016 · doi:10.1017/S0305004100051872
[22] V. I. Averbukh and O. G. Smolyanov, ”Differentiation theory in linear topological spaces,” Usp. Math. Nauk, 22, No. 6, 201–260 (1967). · Zbl 0195.42601
[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). · Zbl 0196.15702
[24] A. Banyaga, ”On the group of equivariant diffeomorphisms,” Topology, 16, 279–283 (1977). · Zbl 0359.57032 · doi:10.1016/0040-9383(77)90009-X
[25] A. Banyaga, ”Sur la structure du groupe des diffeomorphismes qui preservent une forme symplectique,” Comment. Math. Helvet., 53, 174–227 (1978). · Zbl 0393.58007 · doi:10.1007/BF02566074
[26] A. Banyaga, ”The group of diffeomorphisms preserving a regular contact form,” Monogr. Enseign. Math., 26, 47–53 (1978). · Zbl 0392.53022
[27] A. Banyaga and J. Pulido, ”On the group of contact diffeomorphisms of \(\mathbb{R}\)2n+1,” Bort. Soc. Matem. Mexicana, 23, No. 2, 43–47 (1978). · Zbl 0485.53036
[28] A. Banyaga, ”On fixed points of symplectic maps,” Invent Math., 56, 215–229 (1980). · Zbl 0446.58008 · doi:10.1007/BF01390045
[29] A. Banyaga, ”Sur la cohomologie du groupe des diffeomorphismes,” C. R. Acad.Sci. Paris, Ser. I, 294, 625–627 (1982). · Zbl 0513.58016
[30] A. Banyaga, ”On isomorphic classical diffeomorphism groups, II,” J. Differ. Geom., 28, No. 1, 23–35 (1988). · Zbl 0632.53035
[31] A. Banyaga, ”Sur la groupe des diffeomorphismes symplectiques,” Lect. Notes. Math., 484, 50–56 (1975). · doi:10.1007/BFb0082145
[32] A. Banyaga, The Structure of Classical Diffeomorphisms Groups, Kluwer Academic Publ., Amsterdam (1997). · Zbl 0874.58005
[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). · Zbl 0739.58066
[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). · Zbl 0476.76003
[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). · Zbl 0696.58012
[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). · Zbl 0977.57039 · doi:10.1090/S0002-9947-01-02808-2
[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). · Zbl 0194.53103
[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). · Zbl 0761.58010 · doi:10.1007/BF01896972
[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). · Zbl 0837.58028 · doi:10.1090/S0894-0347-1995-1303026-5
[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). · Zbl 0319.53026
[46] W. M. Boothby, ”The transitivity of the automorphisms of certain geometric structures,” Trans. Amer. Math. Soc., 137, 93–100 (1969). · Zbl 0181.49503 · doi:10.1090/S0002-9947-1969-0236961-0
[47] R. Bott, ”On the characteristic classes of group of diffeomorphisms,” Monogr. Enseign. Math., 26, 63–74 (1978).
[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). · Zbl 0919.57026 · doi:10.1155/S1073792898000038
[49] R. Brooks, ”Volumes and characteristic classes of foliations,” Topology, 18, 295–304 (1979). · Zbl 0436.57012 · doi:10.1016/0040-9383(79)90020-X
[50] U. Bunke, Higher analytic torsion and cohomology of diffeomorphism groups, E-print dg-ga/9712001 (1997), http://xxx.lanl.gov.
[51] U. Bunke, ”Higher analytic torsion of sphere bundles and continuous cohomology of Diff(S 2n),” E-print math.DG/9802100 (1998), http://xxx.lanl.gov.
[52] D. Burghelea, ”On the homotopy type of Diff(M n ) and connected problems,” Colloq. Int. CNRS, 7, No. 210, 3–17 (1973). · Zbl 0258.57004
[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. · Zbl 0209.25801
[54] J. Cheeger and J. Simons, ”Differential characters and geometric invariants,” Lect. Notes Math., 1167, 50–80 (1985). · Zbl 0621.57010 · doi:10.1007/BFb0075216
[55] S. Chern and J. Simons, ”Characteristic forms and geometric invariants,” Ann. Math., 99, No. 1, 48–69 (1974). · Zbl 0283.53036 · doi:10.2307/1971013
[56] P. R. Chernnoff, ”Irreducible representations of infinite-dimensional transformation groups and Lie algebras,” Bull. Amer. Math. Soc., 13, No. 1, 46–48 (1985). · Zbl 0578.58022 · doi:10.1090/S0273-0979-1985-15359-5
[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). · Zbl 1039.37068 · doi:10.1088/0305-4470/35/32/201
[59] A. Constantin and B. Kolev, ”Geodesic flow on the diffeomorphism group of the circle,” Comment. Math. Helv., 78, 787–804 (2003). · Zbl 1037.37032 · doi:10.1007/s00014-003-0785-6
[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). · Zbl 1060.58008 · doi:10.1023/A:1014706404871
[61] A. A. Dezin, ”Invariant forms and some structure properties of the Euler equations of hydrodynamics,” Z. Anal. Anwend., 2, 401–409 (1983). · Zbl 0542.76004
[62] S. K. Donaldson, ”Moment maps and diffeomorphisms,” Asian J. Math., 3, No. 1, 1–16 (1999). · Zbl 0999.53053
[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). · Zbl 0419.57006
[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). · Zbl 0196.09402 · doi:10.1090/S0002-9904-1967-11746-4
[65] C. J. Earle and J. Eells, ”A fibre bundle description of Teichmuller theory,” J. Differ. Geom., 3, 19–43 (1969). · Zbl 0185.32901
[66] D. Ebin, ”The manifold of Riemannian metrics,” Proc. Symp. Pure Math., 15, 11–40 (1970). · Zbl 0205.53702
[67] D. Ebin, ”Integrability of perfect fluid motion,” Commun. Pure Appl. Math., 36, No. 1, 37–54 (1983). · Zbl 0525.76022 · doi:10.1002/cpa.3160360103
[68] D. Ebin and J. Marsden, ”Groups of diffeomorphisms and the motion of an incompressible fluid,” Ann. Math., 92, No. 1, 102–163 (1970). · Zbl 0211.57401 · doi:10.2307/1970699
[69] J. Eells, ”On the geometry of function spaces,” in: Symp. Topology Algebra, Mexico (1958), pp. 303–307. · Zbl 0092.11302
[70] J. Eells, ”On submanifolds of certain function spaces,” Proc. Natl. Acad. Sci., 45, No. 10, 1520–1522 (1959). · Zbl 0123.16301
[71] J. Eells, ”Alexander-Pontryagin duality in function spaces,” Proc. Symp. Pure Math., 3, 109–129 (1961). · Zbl 0107.16702
[72] J. Eells, ”A setting for global analysis,” Bull. Amer. Math. Soc., 72, 751–787 (1966). · doi:10.1090/S0002-9904-1966-11558-6
[73] J. Eichhorn, ”The manifold structure of maps between open manifolds,” Ann. Global Anal. Geom., 11, 253–300 (1993). · Zbl 0840.58014
[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). · Zbl 0833.58008
[76] J. Eichhorn and R. Schmid, ”Form preserving diffeomorphisms on open manifolds,” Ann. Global Anal. Geom., 14, 147–176 (1996). · Zbl 0862.58007
[77] J. Eichhorn, ”Diffeomorphism groups on noncompact manifolds,” Zap. Nauch. Semin. POMI, 234, 41–64 (1996). · Zbl 0930.58007
[78] J. Eichhorn and J. Fricke, ”The module structure theorem for Sobolev spaces on open manifolds,” Math. Nachr., 184, 35–47 (1998). · Zbl 0954.46020
[79] J. Eichhorn, ”Poincaré’s theorem and Teichmuller theory for open manifolds,” Asian J. Math., 2, No. 2, 355–404 (1998). · Zbl 1045.58006
[80] J. Eichhorn, ”A classification approach for open manifolds,” Zap. Nauchn. Semin. POMI, 267, 9–45 (2000). · Zbl 1058.53025
[81] Ya. Eliashberg and L. Polterovich, ”Bi-invariant metrics on the group of Hamiltinian diffeomorphisms,” Int. J. Math., 4, No. 5, 727–738 (1993). · Zbl 0795.58016 · doi:10.1142/S0129167X93000352
[82] Ya. Eliashberg and T. Ratiu, ”The diameter of the symplectomorphism group is infinite,” Invent. Math., 103, No. 2, 327–340 (1991). · Zbl 0725.58006
[83] H. Eliasson, ”On the geometry of manifold of maps,” J. Differ. Geom., 1, 169–194 (1967). · Zbl 0163.43901
[84] D. B. A. Epstein, ”The simplicity of certain groups of homeomorphisms,” Compos. Math., 22, 165–173 (1970). · Zbl 0205.28201
[85] D. B. A. Epstein, ”Commutators of C diffeomorphisms,” Comment. Math. Helv., 59, 111–122 (1984). · Zbl 0535.58007 · doi:10.1007/BF02566339
[86] J. Etnyre and R. Ghrist, Contact topology and hydrodynamics, II: Solid tori, E-print math.SG/9907112 (1999), http://xxx.lanl.gov. · Zbl 1098.76011
[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. · Zbl 0979.53018
[89] R. P. Filipkewicz, ”Isomorphisms between diffeomorphism groups,” Ergodic Theor. Dynam. Syst., 2, 159–171 (1982). · Zbl 0521.58016
[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). · Zbl 0532.32008 · doi:10.1007/BF01456093
[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). · doi:10.1016/0370-2693(88)90220-1
[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). · Zbl 0592.17011
[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. · Zbl 0653.17008
[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). · Zbl 0302.57008
[96] K. Fukui, ”Homologies of the group of Diff\(\mathbb{R}\)n, 0),” J. Math. Kyoto Univ., 20, 475–487 (1980). · Zbl 0476.57016
[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). · Zbl 0176.11501
[98] V. L. Ginzburg, ”Some remarks on symplectic actions of compact groups,” Math. Z., 210, 625–640 (1992). · Zbl 0759.57023 · doi:10.1007/BF02571819
[99] Yu. E. Gliklikh, Analysis on Riemannian Manifolds and Problems of Mathematical Physics [in Russian], VGU, Voronezh (1989). · Zbl 0681.53058
[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). · Zbl 0294.58004
[102] J. Grabowski, ”Free subgroups of diffeomorphisms groups,” Fundam. Math., 131, No. 2, 103–121 (1988). · Zbl 0666.58011
[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). · Zbl 0265.58002
[104] R. E. Greene and K. Shiohama, ”Diffeomorphisms and volume-preserving embeddings of noncompact manifolds,” Trans. Amer. Math. Soc., 255, 403–414 (1979). · Zbl 0418.58002 · doi:10.1090/S0002-9947-1979-0542888-3
[105] M. Gromov, ”Pseudoholomorphic curves in symplectic manifolds,” Invent. Math., 82, No. 2, 307–347 (1985). · Zbl 0592.53025 · doi:10.1007/BF01388806
[106] M. Gromov, Partial Differential Relations [Russian translation], Mir, Moscow (1990). · Zbl 0748.47012
[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). · Zbl 0223.17007
[110] S. Haller and T. Rybicki, On the perfectness of nontransitive groups of diffeomorphisms, E-print math.DG/9902095 (1999), http://xxx.lanl.gov. · Zbl 0940.53044
[111] R. S. Hamilton, ”The inverse function theorem of Nash and Moser,” Bull. Amer. Math. Soc., 7, No. 1, 65–222 (1982). · Zbl 0499.58003 · doi:10.1090/S0273-0979-1982-15004-2
[112] D. Hart, ”On the smoothness of generators,” Topology, 22, No. 3, 357–363 (1983). · Zbl 0525.57020 · doi:10.1016/0040-9383(83)90021-6
[113] Y. Hatakeyama, ”Some notes on the groups of automorphisms of contact and symplectic structures,” Tohoku Math. J., 18, 338–347 (1966). · Zbl 0173.24301 · doi:10.2748/tmj/1178243425
[114] A. Hatcher, ”A proof of the Smale conjecture Diff(S 3) O(4),” Ann. Math., 117, 553–607 (1983). · Zbl 0531.57028 · doi:10.2307/2007035
[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. · Zbl 0885.57008
[116] Y. Hattori, ”Ideal magnetohydrodynamics and passive scalar motion as geodesics on semidirect product groups,” J. Phys. A: Math. Gen., 27, L21–L25 (1994). · Zbl 0824.76091 · doi:10.1088/0305-4470/27/2/004
[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). · Zbl 0217.49602
[118] M. R. Herman, ”Sur la groupe des diffeomorphismes du tore,” Ann. Inst. Fourier, 23, No. 2, 75–86 (1973). · Zbl 0269.58004
[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). · doi:10.1103/PhysRevLett.80.4173
[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). · Zbl 0951.37020 · doi:10.1006/aima.1998.1721
[123] H. Hofer, ”Estimates for the energy of a symplectic map,” Comment. Math. Helv., 68, No. 1, 48–72 (1993). · Zbl 0787.58017 · doi:10.1007/BF02565809
[124] H. Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics, Birkhauser, Basel-Boston-Berlin (1994). · Zbl 0805.58003
[125] J. Hoppe, ”DiffA T 2 and the curvature of some infinite dimensional manifolds,” Phys. Lett. B, 215, No. 4, 706–710 (1988). · doi:10.1016/0370-2693(88)90046-9
[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). · Zbl 0256.58002
[129] R. S. Ismagilov, ”On unitary representations of diffeomorphism groups of the space \(\mathbb{R}\)n, n 2,” Funkts. Anal. Prilozh., 9, 71–72 (1975). · Zbl 0328.32008 · doi:10.1007/BF01078188
[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).
[131] R. S. Ismagilov, ”Unitary representations of the measure-preserving diffeomorphism group,” Funkts. Anal. Prilozh., 1, No. 3, 80–81 (1977). · Zbl 0443.58012
[132] R. S. Ismagilov, ”Inductive limits of the area-preserving diffeomorphism groups,” Funkts. Anal. Prilozh., 37, No. 3, 36–50 (2003). · Zbl 1047.22022
[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). · Zbl 0820.58019 · doi:10.1016/0167-2789(89)90030-4
[136] A. A. Kirillov, Elements of Representation Theory [in Russian], Nauka, Moscow (1972). · Zbl 0249.22012
[137] A. A. Kirillov, ”Infinite-dimensional Lie groups: Their orbits, invariants and representations. The geometry of moments,” Lect. Notes Math., 970, 101–123 (1982). · Zbl 0498.22017 · doi:10.1007/BFb0066026
[138] A. A. Kirillov, ”Kähler structure on K-orbits of diffeomorphisms group of the circle,” Funkts. Anal. Prilozh., 21, No. 2, 42–45 (1987). · Zbl 0653.26012 · doi:10.1007/BF01077984
[139] A. A. Kirillov, ”The orbit method. I: Geometric quantization; II: Infinite-dimensional Lie groups and Lie algebras,” Contemp. Math., 145, 1–63 (1993). · Zbl 0935.22016
[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).
[141] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vols. 1, 2 [Russian translation], Nauka, Moscow (1981). · Zbl 0508.53002
[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). · Zbl 0546.58005 · doi:10.1007/BF01438267
[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). · Zbl 0223.53028 · doi:10.1007/BFb0079068
[145] F. Lalonde and D. McDu., ”The geometry of symplectic energy,” Ann. Math., 141, No. 2, 319–333 (1995). · Zbl 0829.53025 · doi:10.2307/2118524
[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. · Zbl 0974.53062
[147] F. Lalonde, D. McDuff, and L. Polterovich, ”Topological rigidity of Hamiltonian loops and quantum cohomology,” Invent. Math., 135, 369–385 (1999). · Zbl 0907.58004 · doi:10.1007/s002220050289
[148] P. F. Lam, ”Embedding a homeomorphism in a flow subject to differentiability conditions,” in: Topological Dynamics, Benjamin, New York (1968), pp. 319–333. · Zbl 0199.59203
[149] L. D. Landau and E. M. Lifshits, Theoretical Physics, Vol. 3, Quantum Mechanics [in Russian], Nauka, Moscow (1989). · Zbl 0714.70004
[150] S. Lang, Introduction to Differentiable Manifolds, New York (1962). · Zbl 0103.15101
[151] J. Leslie, ”On a differential structure for the group of diffeomorphisms,” Topology, 6, 263–271 (1967). · Zbl 0147.23601 · doi:10.1016/0040-9383(67)90038-9
[152] M. V. Losik, ”On Frech’et spaces as diffeologic spaces,” Izv. Vyssh. Ucheb. Zaved., Ser. Mat., 5, 36–42 (1992). · Zbl 0774.58002
[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). · Zbl 0414.35055 · doi:10.1007/BF01076436
[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). · Zbl 0816.58009 · doi:10.1007/BF00773452
[164] J. N. Mather, ”Commutators of diffeomorphisms,” Comment. Math. Helv., 49, 512–528 (1974). · Zbl 0289.57014 · doi:10.1007/BF02566746
[165] J. N. Mather, ”Commutators of diffeomorphisms, II,” Comment. Math. Helv., 50, 33–40 (1975). · Zbl 0299.58008 · doi:10.1007/BF02565731
[166] J. N. Mather, ”A curious remark concerning the geometric transfer map,” Comment. Math. Helv., 59, 86–110 (1984). · Zbl 0535.58006 · doi:10.1007/BF02566338
[167] J. N. Mather, ”Commutators of diffeomorphisms, III,” Comment. Math. Helv., 60, No. 1, 122–124 (1985). · Zbl 0575.58011 · doi:10.1007/BF02567403
[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). · doi:10.1112/jlms/s2-18.2.353
[169] D. McDuff, ”The homology of some groups of diffeomorphisms,” Comment. Math. Helv., 55, 97–120 (1980). · Zbl 0448.57015 · doi:10.1007/BF02566677
[170] D. McDuff, ”Local homology of groups of volume-preserving diffeomorphisms, I,” Ann. Sci. Éc. Norm. Super., 15, 609–648 (1982). · Zbl 0577.58005
[171] D. McDuff, ”Local homology of groups of volume-preserving diffeomorphisms, II,” Comment. Math. Helv., 58, 135–165 (1983). · Zbl 0598.57020 · doi:10.1007/BF02564630
[172] D. McDuff, ”Local homology of groups of volume-preserving diffeomorphisms, III,” Ann. Sci. Éc. Norm. Super., Ser. 4, 16, 529–540 (1983). · Zbl 0619.58008
[173] D. McDuff, ”Some canonical cohomology classes on groups of volume preserving diffeomorphisms,” Trans. Amer. Math. Soc., 275, No. 1, 345–356 (1983). · Zbl 0522.57029 · doi:10.1090/S0002-9947-1983-0678355-7
[174] D. McDuff, ”Symplectic diffeomorphisms and the flux homomorphism,” Invent. Math., 77, No. 2, 353–366 (1984). · Zbl 0546.58014 · doi:10.1007/BF01388450
[175] D. McDuff, ”Remarks on the homotopy type of groups of symplectic diffeomorphisms,” Proc. Amer. Math. Soc., 94, No. 2, 348–352 (1985). · Zbl 0569.57020 · doi:10.1090/S0002-9939-1985-0784191-0
[176] D. McDuff, ”The moment map for circle actions on symplectic manifolds,” J. Geom. Phys., 5, 149–161 (1988). · Zbl 0696.53023 · doi:10.1016/0393-0440(88)90001-0
[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. · Zbl 1102.57013
[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. · Zbl 0284.58002
[180] J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, Springer-Verlag (1999). · Zbl 0933.70003
[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). · Zbl 0979.58004 · doi:10.1007/PL00001631
[182] J. Marsden, T. Ratiu, and A. Weinstein, ”Semidirect products and reduction in mechanics,” Trans. Amer. Math. Soc., 281, No. 1, 147–177 (1984). · Zbl 0529.58011 · doi:10.1090/S0002-9947-1984-0719663-1
[183] J. Marsden and A. Weinstein, ”The Hamiltonian structure of the Maxwell-Vlasov equations,” Phys. D, 4, 394–406 (1982). · Zbl 1194.35463 · doi:10.1016/0167-2789(82)90043-4
[184] J. E. Marsden and A. Weinstein, ”Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids,” Phys. D, 7, 305–323 (1983). · Zbl 0576.58008 · doi:10.1016/0167-2789(83)90134-3
[185] J. N. Mather, ”Simplicity of certain groups of diffeomorphisms,” Bull. Amer. Math. Soc., 80, No. 2, 211–273 (1974). · Zbl 0275.58007 · doi:10.1090/S0002-9904-1974-13456-7
[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). · Zbl 0634.57015
[187] W. Michor and C. Vizman, ”n-Transitivity of certain diffeomorphism groups,” Acta Math. Univ. Comenianae, 63, No. 2, 1–4 (1994). · Zbl 0824.58014
[188] J. W. Milnor, ”On spaces having the homotopy type of a CW complex,” Trans. Amer. Math. Soc., 90, 272–280 (1959). · Zbl 0084.39002
[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. · Zbl 0594.22009
[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). · Zbl 0383.58006
[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). · Zbl 0799.58019 · doi:10.1512/iumj.1993.42.42011
[192] G. Misiolek, ”Conjugate points in \(\mathcal{D}_\mu (T^2 )\) ,” Proc. Amer. Math. Soc., 124, 977–982 (1996). · Zbl 0849.58004 · doi:10.1090/S0002-9939-96-03149-8
[193] G. Misiolek, ”A shallow water equation as a geodesic flow on the Bott-Virasoro group,” J. Geom. Phys., 24, 203–208 (1998). · Zbl 0901.58022 · doi:10.1016/S0393-0440(97)00010-7
[194] G. Misiolek, ”The exponential map on the free loop spaces is Fredholm,” Geom. Funct. Anal., 7, 954–969 (1997). · Zbl 0906.58002 · doi:10.1007/s000390050032
[195] D. Montgomery and L. Zippin, Transformation Groups, Interscience, New York (1955). · Zbl 0068.01904
[196] T. Morimoto and N. Tanaka, ”The classification of real primitive infinite Lie algebras,” J. Math. Kyoto Univ., 10, 207–243 (1970). · Zbl 0211.05401
[197] J. Moser, ”On the volume elements on a manifold,” Trans. Amer. Math. Soc., 120, 286–294 (1965). · Zbl 0141.19407 · doi:10.1090/S0002-9947-1965-0182927-5
[198] S. Nag and A. Verjovsky, ”Diff(S 1) and the Teichmuller spaces,” Commun. Math. Phys., 130, No. 1, 123–138 (1990). · Zbl 0705.32013 · doi:10.1007/BF02099878
[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). · Zbl 0748.76026 · doi:10.1088/0305-4470/25/2/003
[200] N. Nakanishi, ”On the structure of infinite transitive primitive Lie algebras,” Proc. Jpn. Acad., 52, 14–16 (1976). · Zbl 0346.17017 · doi:10.3792/pja/1195518416
[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). · Zbl 0246.58012
[203] H. Omori, ”On the group of diffeomorphisms on a compact manifold,” Proc. Symp. Pure Math., 15, 167–183 (1970). · Zbl 0214.48805
[204] H. Omori, ”Local structures of groups of diffeomorphisms,” J. Math. Soc. Jpn., 24, No. 1, 60–88 (1972). · Zbl 0225.58004 · doi:10.2969/jmsj/02410060
[205] H. Omori, ”On smooth extension theorems,” J. Math. Soc. Jpn., 24, No. 3, 405–432 (1972). · Zbl 0235.58003 · doi:10.2969/jmsj/02430405
[206] H. Omori, ”Group of diffeomorphisms and their subgroups,” Trans. Amer. Math. Soc., 179, 85–122 (1973). · Zbl 0269.58005 · doi:10.1090/S0002-9947-1973-0377975-0
[207] H. Omori, Infinite-Dimensional Lie Transformations Groups, Lect. Notes Math., 427 (1974). · Zbl 0328.58005
[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). · Zbl 0271.58006
[209] K. Ono, ”Some remarks on group actions in symplectic geometry,” J. Fac. Sci. Univ. Tokyo, Sec. IA, 35, 431–437 (1988). · Zbl 0711.53025
[210] K. Ono, ”Equivariant projective imbeddings theorem for symplectic manifolds,” J. Fac. Sci. Univ. Tokyo, Sec. IA, 35, 381–392 (1988). · Zbl 0655.53027
[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). · Zbl 0779.76103 · doi:10.1007/BF01480692
[212] R. Palais, ”Homotopy theory of infinite-dimensional manifolds,” Topology, 5, 1–16 (1966). · Zbl 0138.18302 · doi:10.1016/0040-9383(66)90002-4
[213] R. Palais, Foundations of Global Nonlinear Analysis, Benjamin, New York (1968). · Zbl 0164.11102
[214] R. Palais, Seminar on the Atiyah-Singer Index Theorem [Russian translation], Mir, Moscow (1970). · Zbl 0202.23103
[215] R. Palais and T. E. Stewart, ”The cohomology of differentiable transformation groups,” Amer. J. Math., 83, No. 4, 623–644 (1961). · Zbl 0104.17703 · doi:10.2307/2372901
[216] J. Palis, ”Vector fields generate few diffeomorphisms,” Bull. Amer. Math. Soc., 80, No. 3, 503–505 (1974). · Zbl 0296.57008 · doi:10.1090/S0002-9904-1974-13470-1
[217] J. Palis and J. C. Yoccoz, ”Rigidity of centralizers of diffeomorphisms,” Ann. Sci. Éc. Norm. Super., Ser. 4, 22, 81–98 (1989). · Zbl 0709.58022
[218] M. A. Parinov, ”On the groups of diffeomorphism preserving nondegenerate analytic covector fields,” Mat. Sb., 186, No. 5, 115–126 (1995). · Zbl 0835.53018
[219] J. F. Plante, ”Diffeomorphisms without periodic points,” Proc. Amer. Math. Soc., 88, 716–718 (1983). · Zbl 0519.58038 · doi:10.1090/S0002-9939-1983-0702306-5
[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). · Zbl 0451.58011 · doi:10.1007/BF01214340
[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). · Zbl 0929.57024 · doi:10.1007/s000290050046
[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). · Zbl 0755.76026
[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). · Zbl 0981.53069
[226] E. Shavgulidze ”Quasi-invariant measures on diffeomorphism groups,” Tr. Mat. Inst. Ross. Akad. Nauk, 217, 189–208 (1997). · Zbl 0916.58008
[227] E. V. Shchepin, ”Hausdorff dimension and dynamics of diffeomorphisms,” Mat. Zametki, 65, No. 3, 457–463 (1999). · Zbl 0962.37008
[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). · Zbl 0598.76006
[231] H. Shimomura, ”Quasi-invariant measures on the group of diffeomorphisms and smooth vectors of unitary representations,” J. Funct. Anal., 187, 406–441 (2001). · Zbl 0997.58004 · doi:10.1006/jfan.2001.3807
[232] S. Shkoller, ”Geometry and curvature of diffeomorphism groups with H 1 metric and mean hydrodynamics,” J. Funct. Anal., 160, 337–365 (1998). · Zbl 0933.58010 · doi:10.1006/jfan.1998.3335
[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). · Zbl 0244.17014
[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). · Zbl 0784.57018 · doi:10.1007/BF01895690
[237] A. Shnirelman, ”Generalized fluid flows, their approximation and applications,” Geom. Funct. Anal., 4, No. 5, 586–620 (1994). · Zbl 0851.76003 · doi:10.1007/BF01896409
[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). · Zbl 0874.93055 · doi:10.1353/ajm.1997.0019
[239] I. M. Singer and S. Sternberg, ”On the infinite groups of Lie and Cartan, I,” J. Anal. Math., 15, 1–114 (1965). · Zbl 0277.58008 · doi:10.1007/BF02787690
[240] S. Smale, ”Diffeomorphisms of the 2-sphere,” Proc. Amer. Math. Soc., 10, 621–626 (1959). · Zbl 0118.39103
[241] S. Smale, ”A survey of some recent developements in differential topology,” Bull. Amer. Math. Soc., 69, 131–185 (1963). · doi:10.1090/S0002-9904-1963-10901-5
[242] S. Smale, ”Differentiable dynamics systems,” Bull. Amer. Math. Soc., 73, 747–817 (1967). · Zbl 0202.55202 · doi:10.1090/S0002-9904-1967-11798-1
[243] S. Smale, ”Topology and mechanics,” Usp. Mat. Nauk, 27, No. 2, 77–133 (1972). · Zbl 0233.57018
[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). · Zbl 0477.58007
[246] N. K. Smolentsev, ”On a certain weak Riemannian structure on the diffeomorphism group,” Izv. Vyssh. Ucheb. Zaved., Ser. Mat., 5, 78–80 (1979). · Zbl 0508.58009
[247] N. K. Smolentsev, ”Integrals of ideal barotropic fluid flows,” Sib. Math. Zh., 23, No. 1, 205–208 (1982). · Zbl 0513.58038
[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). · Zbl 0552.58007
[250] N. K. Smolentsev, ”On the vector product on a seven-dimensional manifold,” Sib. Math. Zh., 25, No. 5, 157–167 (1984). · Zbl 0564.53014
[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 \(\frac{\partial }{{\partial t}}\Delta F = \{ \Delta F,F\} \) ,” 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). · Zbl 0778.53036
[255] N. K. Smolentsev, ”Curvature of the classical diffeomorphism groups,” Sib. Mat. Zh., 74, No. 1, 169–176 (1994). · Zbl 0833.53027
[256] S. E. Stepanov and I. G. Shandra, ”Seven classes of harmonic diffeomorphisms,” Mat. Zametki, 74, No. 5, 752–761 (2003). · Zbl 1126.53312
[257] S. E. Stepanov and I. G. Shandra, ”Geometry of infinitesimal harmonic transformations,” Ann. Global Anal. Geom., 24, No. 3, 291–299 (2003). · Zbl 1035.53090 · doi:10.1023/A:1024753028255
[258] S. Sternberg, Lectures on Differential Geometry, Prentice Hall, Englewood Cliffs, New Jersey (1964). · Zbl 0129.13102
[259] F. Takens, ”Characterization of a differentiable structure by its group of diffeomorphisms,” Bol. Soc. Bras. Math., 10, No. 1, 17–26 (1979). · Zbl 0447.58002 · doi:10.1007/BF02588337
[260] W. Thurston, ”Foliations and groups of diffeomorphisms,” Bull. Amer. Math. Soc., 80, No. 2, 04–307 (1974). · Zbl 0295.57014 · doi:10.1090/S0002-9904-1974-13475-0
[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). · Zbl 0337.58003
[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). · Zbl 0383.70020
[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). · Zbl 0593.58036
[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). · Zbl 0655.58009
[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). · Zbl 1035.32009
[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). · Zbl 0890.53060 · doi:10.1016/S0167-2789(96)00192-3
[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). · Zbl 0968.22017
[274] V. Zeitlin and T. Kambe, ”Two-dimensional ideal magnetohydrodynamics and differential geometry,” J. Phys. A: Math. Gen., 26, 5025–5031 (1993). · Zbl 0807.76096 · doi:10.1088/0305-4470/26/19/031
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