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Volumn 69, Issue , 2007, Pages 133-215

Some geometric evolution equations arising as geodesic equations on groups of diffeomorphisms including the Hamiltonian approach

Author keywords

Burgers equation; Connection; Diffeomorphism group; Jacobi field; KdV equation; Symplectic structure

Indexed keywords


EID: 85045532817     PISSN: 14211750     EISSN: 23740280     Source Type: Book Series    
DOI: 10.1007/978-0-8176-4521-2_11     Document Type: Chapter
Times cited : (38)

References (51)
  • 1
    • 0003595232 scopus 로고
    • Geometrical methods in the theory of ordinary differential equations
    • Springer-Verlag, New York
    • V. I. Arnold, Geometrical methods in the theory of ordinary differential equations, Springer-Verlag, New York, 1983. Fundamental Principles of Mathematical Sciences, 250.
    • (1983) Fundamental Principles of Mathematical Sciences , pp. 250
    • Arnold, V.I.1
  • 2
    • 0002771146 scopus 로고
    • An a priori estimate in the theory of hydrodynamic stability
    • (Russian)
    • V.I. Arnold, An a priori estimate in the theory of hydrodynamic stability, Izvestia Vyssh. Uchebn. Zaved. Matematicka 54(1966), 3–5. (Russian).
    • (1966) Izvestia Vyssh. Uchebn. Zaved. Matematicka , vol.54 , pp. 3-5
    • Arnold, V.I.1
  • 3
    • 0001356905 scopus 로고
    • Sur la géometrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits
    • V.I. Arnold, Sur la géometrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits, Ann. Inst. Fourier (Grenoble) 16(1966), 319–361.
    • (1966) Ann. Inst. Fourier (Grenoble) , vol.16 , pp. 319-361
    • Arnold, V.I.1
  • 4
    • 0003319234 scopus 로고    scopus 로고
    • Mathematical methods of classical mechanics, Springer-Verlag, Berlin, Heidelberg, New York, 1978
    • V.I. Arnold, Mathematical methods of classical mechanics, Springer-Verlag, Berlin, Heidelberg, New York, 1978. Graduate Texts in Math. 60.
    • Graduate Texts in Math , vol.60
    • Arnold, V.I.1
  • 6
    • 25444459733 scopus 로고
    • Two natural metrics and their covariant derivatives on a manifold of embeddings
    • E. Binz, Two natural metrics and their covariant derivatives on a manifold of embeddings, Monatsh. Math. 89(1980), 275–288.
    • (1980) Monatsh. Math. , vol.89 , pp. 275-288
    • Binz, E.1
  • 8
    • 0001079851 scopus 로고
    • On the characteristic classes of groups of diffeomorphisms
    • R. Bott, On the characteristic classes of groups of diffeomorphisms, Enseign. Math. 23(1977), 209–220.
    • (1977) Enseign. Math , vol.23 , pp. 209-220
    • Bott, R.1
  • 9
    • 0003988801 scopus 로고    scopus 로고
    • Loop spaces, characteristic classes, and geometric quantization, Birkhäuser, Boston, Basel, Berlin, 1993
    • Jean-Luc Brylinski, Loop spaces, characteristic classes, and geometric quantization, Birkhäuser, Boston, Basel, Berlin, 1993. Progress in Math. 107.
    • Progress in Math , vol.107
    • Brylinski, J.-L.1
  • 11
    • 12044254491 scopus 로고
    • An integrable shallow water equation with peaked solitons
    • R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71(1993), 1661–1664.
    • (1993) Phys. Rev. Lett. , vol.71 , pp. 1661-1664
    • Camassa, R.1    Holm, D.2
  • 13
    • 0242350978 scopus 로고    scopus 로고
    • Geodesic flow on the diffeomorphism group of the circle
    • A. Constantin and B. Kolev, Geodesic flow on the diffeomorphism group of the circle, Comm. Math. Helv. 78(2003), 787–804.
    • (2003) Comm. Math. Helv. , vol.78 , pp. 787-804
    • Constantin, A.1    Kolev, B.2
  • 14
    • 0004034652 scopus 로고    scopus 로고
    • Ordinary differential equations in Banach spaces
    • Springer-Verlag, Berlin, New York
    • K. Deimling, Ordinary differential equations in Banach spaces. Lecture Notes in Mathematics, Vol. 596. Springer-Verlag, Berlin, New York, 1977.
    • Lecture Notes in Mathematics , vol.596 , pp. 1977
    • Deimling, K.1
  • 15
    • 0000950801 scopus 로고
    • The manifold structure of maps between open manifolds
    • J. Eichhorn, The manifold structure of maps between open manifolds, Ann. Global Anal. Geom. 11(1993), 253–300.
    • (1993) Ann. Global Anal. Geom , vol.11 , pp. 253-300
    • Eichhorn, J.1
  • 16
    • 34247543532 scopus 로고    scopus 로고
    • Diffeomorphism groups on noncompact manifolds
    • Differ. Geom. Gruppy Li i Mekh., 15-1, 41–64, 262; translation in J. Math. Sci. (New York) 94(1999), 1162–1176
    • J. Eichhorn, Diffeomorphism groups on noncompact manifolds, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 234(1996), Differ. Geom. Gruppy Li i Mekh., 15-1, 41–64, 262; translation in J. Math. Sci. (New York) 94(1999), 1162–1176.
    • (1996) Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) , vol.234
    • Eichhorn, J.1
  • 17
    • 0032429798 scopus 로고    scopus 로고
    • The module structure theorem for Sobolev spaces on open manifolds
    • J. Eichhorn and J. Fricke, The module structure theorem for Sobolev spaces on open manifolds, Math. Nachr. 194(1998), 35–47.
    • (1998) Math. Nachr. , vol.194 , pp. 35-47
    • Eichhorn, J.1    Fricke, J.2
  • 18
    • 0010862863 scopus 로고
    • The simplicity of certain groups of homeomorphisms
    • D.B.A. Epstein, The simplicity of certain groups of homeomorphisms, Com-positio Math. 22(1970), 165–173.
    • (1970) Com-Positio Math , vol.22 , pp. 165-173
    • Epstein, D.B.A.1
  • 19
    • 0001666357 scopus 로고
    • Note sur une nouvelle formule du calcul différentiel
    • C.F. Faà di Bruno, Note sur une nouvelle formule du calcul différentiel, Quart. J. Math. 1(1855), 359–360.
    • (1855) Quart. J. Math , vol.1 , pp. 359-360
    • Faà Di Bruno, C.F.1
  • 21
    • 85045524866 scopus 로고    scopus 로고
    • Cohomology of infinite dimensional Lie algebras. Nauka, Moscow, 1984 (Russian)
    • Plenum Press), New York
    • D. Fuks, Cohomology of infinite dimensional Lie algebras. Nauka, Moscow, 1984 (Russian). Transl. English, Contemporary Soviet Mathematics. Consultants Bureau (Plenum Press), New York, 1986.
    • Transl. English, Contemporary Soviet Mathematics. Consultants Bureau , pp. 1986
    • Fuks, D.1
  • 22
  • 23
    • 0000891627 scopus 로고
    • Cohomologies of the Lie algebra of vector fields on the circle,(Russian) Funktsional
    • I. M. Gelfand and D. B. Fuks, Cohomologies of the Lie algebra of vector fields on the circle,(Russian) Funktsional. Anal. i Prilozhen. 2(1968), 92–93.
    • (1968) Anal. I Prilozhen. , vol.2 , pp. 92-93
    • Gelfand, I.M.1    Fuks, D.B.2
  • 24
    • 85045530554 scopus 로고    scopus 로고
    • The Analysis of linear partial differential operators I, II, III, IV. Springer-Verlag, Berlin, Heidelberg, New York, 1983, 1983, 1985, 1985
    • 257, 274, 275
    • L. Hörmander, The Analysis of linear partial differential operators I, II, III, IV. Springer-Verlag, Berlin, Heidelberg, New York, 1983, 1983, 1985, 1985. Grundlehren 256, 257, 274, 275.
    • Grundlehren , vol.256
    • Hörmander, L.1
  • 25
    • 25444492413 scopus 로고
    • A note on the manifold of immersions and its Riemannian curvature
    • G. Kainz, A note on the manifold of immersions and its Riemannian curvature, Monatsh. Math. 98(1984), 211–217.
    • (1984) Monatsh. Math , vol.98 , pp. 211-217
    • Kainz, G.1
  • 27
    • 0001014968 scopus 로고
    • The orbits of the group of diffeomorphisms of the circle, and local Lie superalgebras
    • A.A. Kirillov, The orbits of the group of diffeomorphisms of the circle, and local Lie superalgebras, (Russian) Funktsional. Anal. i Prilozhen. 15(1981), 75– 76.
    • (1981) Russian) Funktsional. Anal. I Prilozhen. , vol.15 , pp. 75-76
    • Kirillov, A.A.1
  • 29
    • 0033483085 scopus 로고    scopus 로고
    • The Camassa–Holm equation on as a geodesic flow on the diffeomorphism group
    • S. Kouranbaeva, The Camassa–Holm equation on as a geodesic flow on the diffeomorphism group, J. Math. Phys. 40(1999), 857–868.
    • (1999) J. Math. Phys. , vol.40 , pp. 857-868
    • Kouranbaeva, S.1
  • 30
    • 0000770764 scopus 로고
    • A convenient setting for real analytic mappings
    • A. Kriegl and P. W. Michor, A convenient setting for real analytic mappings, Acta Math. 165(1990), 105–159.
    • (1990) Acta Math , vol.165 , pp. 105-159
    • Kriegl, A.1    Michor, P.W.2
  • 32
    • 21744455992 scopus 로고    scopus 로고
    • Regular infinite dimensional Lie groups
    • A. Kriegl and P. W. Michor, Regular infinite dimensional Lie groups, J. Lie Theory 7(1997), 61–99.
    • (1997) J. Lie Theory , vol.7 , pp. 61-99
    • Kriegl, A.1    Michor, P.W.2
  • 33
    • 0002357489 scopus 로고
    • The generic property of existence of solutions of differential equations in Banach space
    • A. Lasota and J. A. Yorke, The generic property of existence of solutions of differential equations in Banach space, J. Differential Equations 13(1973), 1–12.
    • (1973) J. Differential Equations , vol.13 , pp. 1-12
    • Lasota, A.1    Yorke, J.A.2
  • 35
    • 51249188609 scopus 로고
    • Commutators of diffeomorphisms I, II, Comment
    • 50 (1975), 33–40
    • J. N. Mather, Commutators of diffeomorphisms I, II, Comment. Math. Helv., 49 (1974), 512–528; 50 (1975), 33–40.
    • (1974) Math. Helv. , vol.49 , pp. 512-528
    • Mather, J.N.1
  • 36
    • 25444495977 scopus 로고
    • Manifolds of smooth maps III: The principal bundle of embeddings of a non-compact smooth manifold
    • P. W. Michor, Manifolds of smooth maps III: The principal bundle of embeddings of a non-compact smooth manifold, Cah. Topol. Géom. Différ. Catég., 21(1980), 325–337.
    • (1980) Cah. Topol. Géom. Différ. Catég. , vol.21 , pp. 325-337
    • Michor, P.W.1
  • 39
    • 31644434690 scopus 로고    scopus 로고
    • Riemannian geometries on spaces of plane curves
    • P. W. Michor and D. Mumford, Riemannian geometries on spaces of plane curves, J. Eur. Math. Soc. 8(2006), 1–48.
    • (2006) J. Eur. Math. Soc. , vol.8 , pp. 1-48
    • Michor, P.W.1    Mumford, D.2
  • 40
    • 29244431608 scopus 로고    scopus 로고
    • Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms
    • P. W. Michor and D. Mumford, Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms, Doc. Math. 10(2005), 217–245.
    • (2005) Doc. Math , vol.10 , pp. 217-245
    • Michor, P.W.1    Mumford, D.2
  • 41
    • 22444453467 scopus 로고    scopus 로고
    • Geometry of the Virasoro–Bott group
    • P. W. Michor and T. Ratiu, Geometry of the Virasoro–Bott group, J. Lie Theory 8(1998), 293–309.
    • (1998) J. Lie Theory , vol.8 , pp. 293-309
    • Michor, P.W.1    Ratiu, T.2
  • 42
    • 0039072121 scopus 로고    scopus 로고
    • Conjugate points in the Bott–Virasoro group and the KdV equation
    • G. Misio̷lek, Conjugate points in the Bott–Virasoro group and the KdV equation, Proc. Amer. Math. Soc. 125(1997), 935–940.
    • (1997) Proc. Amer. Math. Soc. , vol.125 , pp. 935-940
    • Misio̷lek, G.1
  • 43
    • 0036436154 scopus 로고    scopus 로고
    • Classical solutions of the periodic Camassa–Holm equation
    • G. Misio̷lek, Classical solutions of the periodic Camassa–Holm equation, Geom. Funct. Anal. 12(2002), 1080–1104.
    • (2002) Geom. Funct. Anal. , vol.12 , pp. 1080-1104
    • Misio̷lek, G.1
  • 44
    • 0031989845 scopus 로고    scopus 로고
    • A shallow water equation as a geodesic flow on the Bott–Virasoro group
    • G. Misio̷lek, A shallow water equation as a geodesic flow on the Bott–Virasoro group, J. Geom. Phys. 24(1998), 203–208.
    • (1998) J. Geom. Phys. , vol.24 , pp. 203-208
    • Misio̷lek, G.1
  • 48
    • 0000896226 scopus 로고
    • The motion of the free n-dimensional rigid body
    • T. Ratiu, The motion of the free n-dimensional rigid body, Indiana Univ. Math. J. 29(1980), 609–629.
    • (1980) Indiana Univ. Math. J , vol.29 , pp. 609-629
    • Ratiu, T.1
  • 49
    • 0001120168 scopus 로고
    • The geometry of the KdV equation, Topological methods in quantum field theory (Trieste, 1990)
    • G. Segal, The geometry of the KdV equation, Topological methods in quantum field theory (Trieste, 1990). Internat. J. Modern Phys. A 6(1991), 2859–2869.
    • (1991) Internat. J. Modern Phys. A , vol.6 , pp. 2859-2869
    • Segal, G.1
  • 50
    • 0002374631 scopus 로고
    • Foliations and groups of diffeomorphisms
    • W. Thurston, Foliations and groups of diffeomorphisms, Bull. Amer. Math. Soc. 80(1974), 304–307.
    • (1974) Bull. Amer. Math. Soc. , vol.80 , pp. 304-307
    • Thurston, W.1
  • 51
    • 0000595516 scopus 로고
    • Symplectic manifolds and their Lagrangian manifolds
    • A. Weinstein, Symplectic manifolds and their Lagrangian manifolds, Adv. Math. 6(1971), 329–345.
    • (1971) Adv. Math , vol.6 , pp. 329-345
    • Weinstein, A.1


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