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The Lie Group Structure of Diffeomorphism Groups and Invertible Fourier Integral Operators with Applications

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Infinite Dimensional Groups with Applications

Part of the book series: Mathematical Sciences Research Institute Publications ((MSRI,volume 4))

Abstract

This is a survey of basic facts about the differentiable structure of infinite dimensional Lie groups. The groups of diffeomorphisms and of invertible Fourier integral operators on a compact manifold have a structure which is weaker than that of a Lie group in the classical sense. This differentiable structure is called ILH (inverse limit of Hilbert) Lie group. We indicate applications to the well-posedness problem, to hydrodynamics, plasma physics, general relativity, quantum field theory, and completely integrable PDE’s.

Part of this material has been presented as a lecture by Rudolf Schmid at the Conference on Infinite Dimensional Lie Groups, MSRI, Berkeley, May 10–15, 1984.

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References

  • Abarbanel, H.D.I., Holm, D.D., Marsden, J.E. and Ratiu, T.S. [1984], Richardson number criterion for the nonlinear stability of stratified flow, Phys. Rev. Lett. 52, 2352–2355.

    Article  MathSciNet  Google Scholar 

  • Abarbanel, H.D.I., Holm, D.D., Marsden, J.E. and Ratiu, T.S. [1985], Nonlinear stability analysis of stratified ideal fluid equilibria, Transactions Phil. Soc. Cambridge (to appear).

    Google Scholar 

  • Abraham, R. [1961], Lectures of Sraale on differential topology, Mimeographed, Columbia.

    Google Scholar 

  • Abraham, R. and Marsden, J.E. [1978], Foundations of Mechanics. Second Ed., Addison-Wesley.

    Google Scholar 

  • Abraham, R., Marsden, J.E. and Ratiu, T.S. [1983], Manifolds. Tensor

    Google Scholar 

  • Analysis, and Applications, Addison-Wesley, Reading, Mass.

    Google Scholar 

  • Adams, R. [1975], Sobolev Spaces, Academic Press, New York.

    MATH  Google Scholar 

  • Adams, M., Ratiu, T.S. and Schmid, R. [1984], A Lie group structure for Fourier integral operators, MSRI Preprint 049-84-7, Berkeley, California.

    Google Scholar 

  • Adler, M. [1979], On a trace functional for formal pseudo-differential operators and the symplectic structure for Korteweg-deVries type equations, Invent. Math. 50, 219–248.

    Article  MATH  Google Scholar 

  • Arms, J.M. [1981], The structure of the solution set for the Yang-Mills equations, Math. Proc. Camb. Phil. Soc. 90, 361–372.

    Article  MathSciNet  MATH  Google Scholar 

  • Arms, J.M., Marsden, J.E. and Moncrief, V. [1982], The structure of the space of solutions of Einstein’s equations II. Several Killing fields and the Einstein-Yang-Mills equations, Ann. of Phys., 144, 81–106.

    Article  MathSciNet  MATH  Google Scholar 

  • Arnold, V.I. [1965], Conditions for nonlinear stability for plane curvilinear flows of an ideal fluid, Doklady Mat. Nauk 162(5), 773–777.

    Google Scholar 

  • Arnold, V.I. [1966], Sur la géometrie differentielle des groupes de Lie de dimension infinie et ses applications a l’hydrodynamique des fluids parfaits, Ann. Inst. Fourier, Grenoble. 16, 319–361.

    Article  Google Scholar 

  • Arnold, V.I. [1969], An a priori estimate in the theory of hydrodynamic stability, Am. Math. Soc. Transl. 19, 267–269.

    Google Scholar 

  • Arnold, V.I. [1978], Mathematical Methods of Classical Mechanics. Springer Graduate Text in Math 60, Springer, New York.

    MATH  Google Scholar 

  • Atiyah, M.F., Hitchin, N.J. and Singer, I.M. [1978], Self-duality in four-dimensional Riemannian geometry, Proc. Royal Soc. London A. 362, 425–461.

    Article  MathSciNet  MATH  Google Scholar 

  • Binz, E. [1979], Einstein’s evolution equation for the vacuum formulated on the space of differentials of immersions, Springer Lecture Notes in Math. 1037.

    Google Scholar 

  • Binz, E. [1984], The space of smooth isometric immersions of a compact manifold into an Euclidean space is a Fréchet manifold, C.R. Math. Rep. Acad. Sci. Canada. Vol. VI(5).

    Google Scholar 

  • Binz, E. and Fischer, H.R. [1978], The manifold of embeddings of a closed manifold, Springer Lecture Notes in Physics 139.

    Google Scholar 

  • Bokobza-Haggiag, J. [1969], Opérateurs pseudo-différentiels sur une variété différentiable, Ann. Inst. Fourier. Grenoble, 19, 125–177.

    Article  MathSciNet  MATH  Google Scholar 

  • Bourbaki, N. [1975], Lie Groups and Lie Algebras. Hermann, Paris.

    MATH  Google Scholar 

  • Brill, D., and Deser, S. [1968], Variational methods and positive energy in relativity, Ann. Phys. 50, 548–570.

    Article  Google Scholar 

  • Chernoff, P.R. and Marsden, J.E. [1974], Properties of infinite dimensional Hamiltonian systems. Lecture Notes in Math., Vol. 425, Springer, Berlin.

    MATH  Google Scholar 

  • Choquet-Bruhat, Y. and Marsden, J.E. [1976], Solution of the local mass problem in general relativity, C.R. Acad. Sci. (Paris), 282,(1976), 609–612, Comm. Math. Phys. 51. (976), 283-296.

    MathSciNet  MATH  Google Scholar 

  • Choquet-Bruhat, Y., DeWitt-Morette, C. and Dillard-Bleick, M. [1982], Analysis. Manifolds and Physics. 2nd ed., North Holland, Amsterdam.

    MATH  Google Scholar 

  • Coifman, R.R. and Meyer, Y. [1978], Au-delà des Opérateurs Pseudo-Différentiels, Société math, de France, Astérisque 57.

    Google Scholar 

  • Duff, G.F.D. [1952], Differential forms in manifolds with boundary, Annals of Math., 56, 115–127.

    Article  MathSciNet  MATH  Google Scholar 

  • Duistermaat, J.J. [1973], Fourier Integral Operators, Lecture Notes, Courant Institute of Mathematical Sciences, New York.

    MATH  Google Scholar 

  • Ebin, D.G. [1968], The manifold of Riemannian metrics, Bull. Am. Math. Soc. 74, 1002–1004.

    Article  MathSciNet  Google Scholar 

  • Ebin, D.G. and Marsden, J.E. [1970], Groups of diffeomorphisms and the motion of incompressible fluid, Ann. of Math. 92, 102–163.

    Article  MathSciNet  MATH  Google Scholar 

  • Eells, J. [1958], On the geometry of function spaces, Symposium de Topologia Algebrica, UNAM, Mexico City, 303–307.

    Google Scholar 

  • Fischer, A. [1970], The theory of superspace, in Relativity, ed. M. Carmelli, S. Fickler, and L. Witten, Plenum Press.

    Google Scholar 

  • Fischer, A. and Marsden, J.E. [1972], The Einstein equations of evolution — a geometric approach, J. Math. Phys. 13, 546–568.

    Article  MathSciNet  MATH  Google Scholar 

  • Fischer, A. and Marsden, J.E. [1973], Linearization stability of the Einstein equations, Bull. Am. Math. Soc. 79, 102–163.

    Article  MathSciNet  Google Scholar 

  • Fischer, A. and Marsden, J.E. [1979], The initial value problem and the dynamical formulation of general relativity, General Relativity, An Einstein Centenary Volume, eds. S.W. Hawking and W. Israel, Cambridge, 138-211.

    Google Scholar 

  • Freed, D.S. and Uhlenbeck, K.K. [1984], Instantons and Four-Manifolds, MSRI Berkeley Publications 1, Springer Verlag.

    Google Scholar 

  • Gardner, C.F. [1971], Korteweg-deVries equation and generalizations IV. The Korteweg-deVries equation as a Hamiltonian system, J. Math. Phys. 12, 1548–1551.

    Article  MATH  Google Scholar 

  • Garland, H. [1980], The Arithmetic Theory of Loop Groups. Publ. Math. I.H.E.S. 52, 181–312.

    Google Scholar 

  • Goldin, G.A. and Sharp, D.H. [1983], Particle spin from representations of the diffeomorphism group, Comm. Math. Phys. 92, 217–228.

    Article  MathSciNet  MATH  Google Scholar 

  • Goodman, R. and Wallach, N.R. [1984], Structure and unitary cocycle representations of loop groups and the group of diffeomorphisms of the circle, J. fuer reine und angewandte Math. (Grelle J.), 347, 69–133.

    MathSciNet  MATH  Google Scholar 

  • Guckenheimer, J. and Holmes, P. [1983], Nonlinear Oscillations. Dynamical Systems, and Bifurcations of Vector Fields. Springer, Applied Math. Sciences, Vol. 42.

    Google Scholar 

  • Guillemin, V. and Sternberg, S. [1977], Geometric Asymptotics. American Mathematical Society Survey, Vol. 14, AMS, Providence, RI.

    Google Scholar 

  • Guillemin, V. and Sternberg, S. [1980], The moment map and collective motion, Ann. of Phys. 127, 220–253.

    Article  MathSciNet  MATH  Google Scholar 

  • Guillemin, V. and Sternberg, S. [1984], Symplectic Techniques in Physics. Cambridge University Press, Cambridge.

    MATH  Google Scholar 

  • Gutknecht, J. [1977], Die Struktur auf der Diffeomorphismen-gruppe einer kompakten Mannigfaltigkeit, Ph.D. Thesis, ETH, Zürich.

    Google Scholar 

  • Hamilton, R. [1982], The inverse function theorem of Nash and Moser, Bull. Am. Math. Soc. 7, 65–222.

    Article  MathSciNet  MATH  Google Scholar 

  • Holm, D., Marsden, J.E., Ratiu, T.S. and Weinstein, A. [1985], Nonlinear stability of fluid and plasma systems, Physics Reports.

    Google Scholar 

  • Holmes, P. and Marsden, J.E. [1983], Horseshoes and Arnold diffusion for Hamiltonian systems on Lie groups, Ind. Univ. Math. J. 32, 273–310.

    Article  MathSciNet  MATH  Google Scholar 

  • Hormander, L. [1971], Fourier integral operators I, Acta Math. 127, 79–183.

    Article  MathSciNet  MATH  Google Scholar 

  • Iacob, A. and Sternberg, S. [1980], Coadjoint structures, solitons, and integrability, Lecture Notes in Physics, Vol. 120, Springer.

    Google Scholar 

  • Isenberg, J. and Marsden, J.E. [1982], A slice theorem for the space of solutions of Einstein’s equations, Phys. Rep. 89, 179–222.

    Article  MathSciNet  Google Scholar 

  • Keller, H.H. [1974], Differential Calculus in Locally Convex Spaces. Lecture Notes in Math. 417, Springer, Berlin.

    MATH  Google Scholar 

  • Kostant, B. [1979], The solution to a generalized Toda Lattice and representation theory, Adv. Math. 34, 195–338.

    Article  MathSciNet  MATH  Google Scholar 

  • Kuchar, K. [1976], The dynamics of tensor fields in hyperspace, J. Math. Phys. 17: 777–791, 792-800, 801-820; 18: 1589-1597.

    Article  MathSciNet  Google Scholar 

  • Kumano-go, H. [1981], Pseudo-Differential Operators, MIT Press, Cambridge, Mass.

    Google Scholar 

  • Lang, S. [1972], Differential Manifolds, Addison-Wesley, Reading, Mass.

    MATH  Google Scholar 

  • Lawson, H.B. [1983], The Theory of Gauge Fields in Four Dimensions. CBMS Conference, University of California, Santa Barbara.

    Google Scholar 

  • Leslie, J. [1967], On a differential structure for the group of diffeomorphisms, Topology 6, 263–271.

    Article  MathSciNet  MATH  Google Scholar 

  • Marsden, J.E. [1974], Applications of Global Analysis in Mathematical Physics, Publish or Perish, Berkeley, CA.

    MATH  Google Scholar 

  • Marsden, J.E. [1976], Well-posedness of the equations of a nonhomogeneous perfect fluid, Comm. PDE 1, 215–230.

    Article  MathSciNet  Google Scholar 

  • Marsden, J.E. [1981], Lectures on Geometric Methods in Mathematical Physics, CBMS, Vol. 37, SIAM, Philadelphia.

    Book  Google Scholar 

  • Marsden, J.E. [1985], Chaos in dynamical systems by the Poincaré-Melnikov-Arnold method, Proceedings of the ARO Conference on Dynamics, SIAM (to appear).

    Google Scholar 

  • Marsden, J.E., Ebin, G.D. and Fischer, A. [1972], Diffeomorphism groups, hydrodynamics and relativity, in Proc. 13th Biennial Seminar of Canadian Math. Congress. J.R. Vanstone (ed.), Montreal, 135-279.

    Google Scholar 

  • Marsden, J.E., Ratiu, T.S. and Weinstein, A. [1984a], Semidirect products and reduction in mechanics, Transact. of AMS. 281, 147–177.

    Google Scholar 

  • Marsden, J.E., Ratiu, T.S. and Weinstein, A. [1984b], Reduction and Hamiltonian structures on duals of semidirect product Lie algebras, Cont. Math. AMS. Vol. 28, 55–100.

    Google Scholar 

  • Marsden, J.E. and Weinstein, A. [1982], The Hamiltonian structure of the Maxwell-Vlasov equations, Physica 4P, 394–406.

    MathSciNet  Google Scholar 

  • Marsden, J.E. and Weinstein, A. [1983], Co-adjoint orbits, vortices and Clebsch variables for incompressible fluids, Physica 7D. 305–323.

    MathSciNet  Google Scholar 

  • Marsden, J.E., Weinstein, A., Ratiu, T.S., Schmid, R. and Spencer, R. [1983], Hamiltonian systems with symmetry, coadjoint orbits and plasma physics, Proc. IUTAM-ISIMM Symposium on Modern Developments in Analytical Mechanics. Atti Accad. Sci. Torino, Suppl., Vol. 117, 289–340.

    MathSciNet  Google Scholar 

  • Marsden, J.E., Weinstein, A., Ratiu, T.S., Schmid, R. and Spencer, R. [1985], The Geometry and Dynamics of Fluids and Plasmas (in preparation 1985).

    Google Scholar 

  • Michor, P. [1980], Manifolds of Differentiable Mappings, Shiva Math. Series, No. 3, Kent, UK.

    Google Scholar 

  • Milnor, J. [1983], Remarks on infinite dimensional Lie groups, Proc. Summer School on Quantum Gravity, ed. B. DeWitt, Les Houches.

    Google Scholar 

  • Mitter, P.K. [1980], Geometry of the space of gauge orbits and the Yang-Mills dynamical system, Proc. NATO Advanced Study Institute on Recent Developments in Gauge Theories, Cargese 1979, 265–292, ed. G. T’Hooft et al., Plenum Press, New York.

    Google Scholar 

  • Moncrief, V. [1975], Spacetime symmetries and linearization stability of the Einstein equations, J. Math. Phys. 16, 493–498.

    Article  MathSciNet  MATH  Google Scholar 

  • Moncrief, V. [1977], Gauge symmetries of Yang-Mills fields, Ann. of Phys. 108, 387–400.

    Article  MathSciNet  Google Scholar 

  • Moncrief, V. [1980], Reduction of the Yang-Mills equations, Proc. Differential Geometrical Methods in Mathematical Physics, Salamanca 1979, Lecture Notes in Math. 836, 276–291, Springer, Berlin.

    Google Scholar 

  • Mulase, M. [1984], Cohomological structure of solutions of soliton equations, isospectral deformations of ordinary differential operators and a characterization of Jacobian varieties, Journal of Diff. Geometry 19(2), 403–430.

    MathSciNet  MATH  Google Scholar 

  • Nui, F. [1983], An effective potential for classical Yang-Mills fields as outline for bifurcation on gauge orbit space, Ann. Phys. 149, 179–202.

    Article  Google Scholar 

  • Omori, H. [1970], On the group of diffeomorphisms on a compact manifold, Proc. Symp. Pure Math. 15, 167–184.

    MathSciNet  Google Scholar 

  • Omori, H. [1974], Infinite Dimensional Lie Transformation Groups, Lecture Notes in Math., 427, Springer, Berlin.

    MATH  Google Scholar 

  • Omori, H., Maeda, Y., Yoshika, A., and Kobayashi, O., [1980–83], On regular Frechet-Lie groups I, II, III, IV, V, VI, Tokyo J. Math. 3,(1980), 353–390; 4 (1981), 221-253; 4 (1981), 255-277; 5 (1982), 365-398; 6 (1983), 39-64; 6 (1983), 217-246.

    Article  MathSciNet  MATH  Google Scholar 

  • Palais, R. [1965], Seminar on the Atiyah-Singer Index Theorem. Ann. of Math. Studies, Vol. 57, Princeton University Press, Princeton, NJ.

    Google Scholar 

  • Palais, R. [1968], Foundations of Global Nonlinear Analysis. Addison-Wesley, Reading, Mass.

    Google Scholar 

  • Parker, P.E. and Taubes, C.H. [1982], On Witten’s proof of the positive energy theorem, Comm. Math. Phys. 84, 223–238.

    Article  MathSciNet  MATH  Google Scholar 

  • Pressley, A. and Segal, G. [1984], Loop Groups and Their Representations. Preprint, to appear in Oxford University Press.

    Google Scholar 

  • Ratiu, T.S. [1979], On the smoothness of the time t-map of the KdV equation and the bifurcation of the eigenvalues of Hill’s operator, Lecture Notes in Math. Vol. 755, Springer, Berlin, 1979.

    Google Scholar 

  • Ratiu, T.S. [1981], Euler-Poisson equations on Lie algebras and the N-dimensional heavy rigid body, Proc. Natl. Acad. Sci. USA 78(3), (1981), 1327–1328, and Amer. Journal of Math. 104(2), (1982), 409-448.

    Article  MathSciNet  MATH  Google Scholar 

  • Ratiu, T.S. and Schmid, R. [1981], The differentiable structure of three remarkable diffeomorphism groups, Math. Zeitschr. 177, 81–100.

    Article  MathSciNet  MATH  Google Scholar 

  • Schmid, R. [1978], Die Symplectomorphismen-Gruppe als Frechet-Lie-Gruppe, Thesis, Univ. Zurich.

    Google Scholar 

  • Schmid, R. [1983], The inverse function theorem of Nash and Moser for the Γ-differentiability, to appear in Proc. Convergence Structures and Applications II, Akademie der Wissenschaften der DDR, Berlin.

    Google Scholar 

  • Schoen, R. and Yau, S.T. [1979], Positivity of the total mass of a general space-time, Phys. Rev. Lett. 43, 1457–1459.

    Article  MathSciNet  Google Scholar 

  • Shiota, T. [1985], Characterization of Jacobian varieties in terms of soliton equations, preprint.

    Google Scholar 

  • Singer, I.M. [1978], Some remarks on the Gribov ambiguity, Commun. Math. Phys. 60, 7–12.

    Article  MATH  Google Scholar 

  • Singer, I.M. [1980], The geometry of the orbit space for nonabelian gauge theories, preprint.

    Google Scholar 

  • Spencer, R. and Schmid, R. [1984], Electrostatic normal modes in an unmagnetized homogeneous Coulomb plasma. A Hamiltonian Approach, Phys. Lett. 101A. 485–490.

    Google Scholar 

  • Symes, W. [1980], Systems of Toda type, inverse spectral problems and representation theory, Inventiones Math. 59, 13–51.

    Article  MathSciNet  MATH  Google Scholar 

  • Symes, W. [1980], Hamiltonian group actions and integrable systems, Physica 1D, 339–374.

    MathSciNet  Google Scholar 

  • Taubes, C.H. [1983], Stability in Yang-Mills theories, Commun. Math. Phys. 91, 235–263.

    Article  MathSciNet  MATH  Google Scholar 

  • Taubes, C.H. [1984], Self-dual connections on 4-manifolds with indefinite intersection matrix, Journal of Diff. Geometry 19, 517–560.

    MathSciNet  MATH  Google Scholar 

  • Taylor, M. [1981], Pseudodifferential Operators, Princeton University Press, Princeton, NJ.

    MATH  Google Scholar 

  • Trèves, F. [1980], Introduction to Pseudodifferential and Fourier Integral Operators I, II, Plenum Press, New York, NY.

    Google Scholar 

  • Tromba, A.J. [1976], Almost-Riemannian structures on Banach manifolds, the Morse lemma and the Darboux theorem, Can. J. Math. 28, 640–652.

    Article  MathSciNet  MATH  Google Scholar 

  • Uhlenbeck, K.K. [1982], Connections with Lp bounds on curvature, Commun. Math. Phys. 83, 31–42.

    Article  MathSciNet  MATH  Google Scholar 

  • Weinstein, A. [1971], Symplectic manifolds and their Lagrangian submanifolds, Advances in Math. 6, 329–346.

    Article  MathSciNet  MATH  Google Scholar 

  • Weinstein, A. [1977], Lectures on Symplectic Manifolds, CBMS, Conference Series, Vol. 29, American Mathematical Society, Providence, RI.

    Google Scholar 

  • Weinstein, A. [1983], The local structure of Poisson manifolds, Journal of Diff. Geometry 18(3), 523–557.

    MATH  Google Scholar 

  • Witten, E. [1981], A new proof of the positive energy theorem, Comm. Math. Phys. 80, 381–402.

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Institute for Advanced Study, Princeton, NJ, 08540, USA

    Malcolm Adams

  2. Department of Mathematics, University of Arizona, Tucson, Arizona, 85721, USA

    Tudor Ratiu

  3. Mathematical Sciences Research Institute, Berkeley, USA

    Tudor Ratiu & Rudolf Schmid

  4. Department of Mathematics, Yale University, New Haven, Connecticut, 06520, USA

    Rudolf Schmid

Authors
  1. Malcolm Adams
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  2. Tudor Ratiu
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  3. Rudolf Schmid
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Editors and Affiliations

  1. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

    V. Kac

  2. Mathematical Sciences Research Institute, 1000 Centennial Drive, Berkeley, CA, 94720, USA

    V. Kac

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Adams, M., Ratiu, T., Schmid, R. (1985). The Lie Group Structure of Diffeomorphism Groups and Invertible Fourier Integral Operators with Applications. In: Kac, V. (eds) Infinite Dimensional Groups with Applications. Mathematical Sciences Research Institute Publications, vol 4. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1104-4_1

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