Annals of Global Analysis and Geometry

, Volume 44, Issue 4, pp 529–540 | Cite as

A zoo of diffeomorphism groups on \(\mathbb{R }^{n}\)

Article

Abstract

We consider the groups \({\mathrm{Diff }}_\mathcal{B }(\mathbb{R }^n)\), \({\mathrm{Diff }}_{H^\infty }(\mathbb{R }^n)\), and \({\mathrm{Diff }}_{\mathcal{S }}(\mathbb{R }^n)\) of smooth diffeomorphisms on \(\mathbb{R }^n\) which differ from the identity by a function which is in either \(\mathcal{B }\) (bounded in all derivatives), \(H^\infty = \bigcap _{k\ge 0}H^k\), or \(\mathcal{S }\) (rapidly decreasing). We show that all these groups are smooth regular Lie groups.

Keywords

Diffeomorphism group Infinite dimensional regular Lie group  Convenient calculus 

Mathematics Subject Classification (1991)

58B20 58D15 

References

  1. 1.
    Bauer, M., Bruveris, M., Harms, P., Michor, P.W.: Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group. Ann. Global Anal. Geom. (2012). doi: 10.1007/s10455-012-9353-x
  2. 2.
    Bauer, M., Bruveris, M., Harms, P., Michor, P.W.: Vanishing geodesic distance for the Riemannian metric with geodesic equation the KdV-equation. Ann. Global Anal. Geom. 41(4), 461–472 (2012)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bauer, M., Bruveris, M., Peter W.M.: Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group II (2013). doi: 10.1007/s10455-013-9370-4
  4. 4.
    Bauer, M, Bruveris, M., Peter W.M.: The homogeneous Sobolev metric of order one on diffeomorphism groups on the real line. arXiv:1209.2836 (2012)Google Scholar
  5. 5.
    Faà di Bruno, C.F.: Note sur une nouvelle formule du calcul différentielle. Quart. J. Math. 1, 359–360 (1855)Google Scholar
  6. 6.
    Frölicher A., Kriegl A.: Linear spaces and differentiation theory. Pure and Applied Mathematics (New York). Wiley, Chichester (1988)Google Scholar
  7. 7.
    Gerald A.G: Lectures on diffeomorphism groups in quantum physics. In: Contemporary problems in mathematical physics, pp 3–93. World Science Publication, Hackensack (2004)Google Scholar
  8. 8.
    Kriegl, A., Michor, P.W.: The convenient setting of global analysis, volume 53 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (1997)CrossRefGoogle Scholar
  9. 9.
    Kriegl, A., Michor, P.W.: Regular infinite-dimensional Lie groups. J. Lie Theory 7(1), 61–99 (1997)MathSciNetMATHGoogle Scholar
  10. 10.
    Micheli, M., Michor, P.W., Mumford, D.: Sobolev metrics on diffeomorphism groups and the derived geometry of spaces of submanifolds. Izvestiya: Mathematics, to appear (2012)Google Scholar
  11. 11.
    Michor, P.W.: A convenient setting for differential geometry and global analysis. Cahiers Topologie Géom. Différentielle 25(1), 63–109 (1984)MathSciNetMATHGoogle Scholar
  12. 12.
    Michor, P.W.: A convenient setting for differential geometry and global analysis. II. Cahiers Topologie Géom. Différentielle 25(2), 113–178 (1984)MathSciNetGoogle Scholar
  13. 13.
    Michor, P.W.: Some geometric evolution equations arising as geodesic equations on groups of diffeomorphisms including the Hamiltonian approach. In: Phase space analysis of partial differential equations, vol. 69 of Programme. Nonlinear Differential Equations Appl., pp. 133–215. Birkhäuser Boston, Boston (2006)Google Scholar
  14. 14.
    Michor, P.W.: Topics in differential geometry, volume 93 of Graduate Studies in Mathematics. American Mathematical Society, Providence (2008)Google Scholar
  15. 15.
    Michor, P.W.: Manifolds of differentiable mappings. Shiva Mathematics Series 3, Orpington (1980)Google Scholar
  16. 16.
    Michor, P.W.: Manifolds of smooth maps II: The Lie group of diffeomorphisms of a non compact smooth manifold. Cahiers Topologie Géom. Différentielle 21, 63–86 (1980)MathSciNetMATHGoogle Scholar
  17. 17.
    Milnor, J.: Remarks on infinite-dimensional Lie groups. In: Relativity, groups and topology, II (Les Houches, 1983), pp. 1007–1057. North-Holland, Amsterdam (1984)Google Scholar
  18. 18.
    Mumford, D., Michor, P. W.: On Euler’s equation and ‘EPDiff’. arXiv:1209.6576 (2012)Google Scholar
  19. 19.
    Omori, H., Maeda, Y., Yoshioka, A.: On regular Fréchet Lie groups IV. Definitions and fundamental theorems. Tokyo J. Math. 5, 365–398 (1982)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Omori, H., Maeda, Y., Yoshioka, A.: On regular Fréchet Lie groups V. Several basic properties. Tokyo J. Math. 6, 39–64 (1983)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Schwartz, L.: Théorie des distributions. Nouvelle édition, Hermann, Paris (1966)Google Scholar
  22. 22.
    Vogt, D.: Sequence space representations of spaces of test functions and distributions. In: Functional analysis, holomorphy, and approximation theory (Rio de Janeiro, 1979), volume 83 of Lecture Notes in Pure and Applied Mathematics, pp. 405–443. Dekker, New York (1983)Google Scholar
  23. 23.
    Walter, B.: Weighted diffeomorphism groups of Banach spaces and weighted mapping groups. Diss. Math. (Rozprawy Mat.) 484, 128 (2012)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Fakultät für Mathematik, Universität WienWienAustria
  2. 2.Division of Applied MathematicsBrown UniversityProvidenceUSA

Personalised recommendations