The group of diffeomorphisms of a non-compact manifold is not regular

Jean-Pierre Magnot 1
  • 1 LAREMA - UMR CNRS 6093, Université d’Angers, 2 Boulevard Lavoisier - 49045 Angers cedex 01 and Lycée Jeanne d’Arc, Avenue de grande bretagne, F-63000 , Clermont-Ferrand, France

Abstract

We show that a group of diffeomorphisms D on the open unit interval I, equipped with the topology of uniform convergence on any compact set of the derivatives at any order, is non-regular: the exponential map is not defined for some path of the Lie algebra. This result extends to the group of diffeomorphisms of finite dimensional, non-compact manifold M.

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  • [1] Omori H., Groups of diffeomorphisms and their subgroups, Trans. Amer. Math. Soc., 1973, 179, 85-122

  • [2] Omori H., A remark on nonenlargeable Lie algebras, J. Math. Soc. Japan, 1981, 33(4), 707-710

  • [3] Omori H., Infinite dimensional Lie groups, AMS Translations of Mathematical Monographs, Amer. Math. Soc., Providence, R.I., 1997, 158

  • [4] Kriegl A., Michor P. W., The convenient setting for global analysis, AMS Math. Surveys and Monographs, AMS, Providence, 1997, 53

  • [5] Khesin B., Wendt R., Geometry of infinite dimensional groups, Springer, 2008

  • [6] Magnot J.-P., Difféologie du fibré d’Holonomie en dimension infinie, C. R. Math. Soc. Roy. Can., 2006, 28(4), 121-127

  • [7] Watts J., Diffeologies, differentiable spaces and symplectic geometry, University of Toronto, PhD thesis, 2013, arXiv:1208.3634v1

  • [8] Frölicher A., Kriegl A., Linear spaces and differentiation theory, Wiley series in Pure and Applied Mathematics, Wiley Interscience, 1988

  • [9] Magnot J.-P., Ambrose-Singer theorem on diffeological bundles and complete integrability of the KP equation, Int. J. Geom. Meth. Mod. Phys., 2013, 10(9), DOI: 10.1142/S0219887813500436

  • [10] Hirsch M., Differential topology, Springer, 1997

  • [11] Kriegl A., Michor P. W., Rainer A., An exotic zoo of diffeomorphism groups on Rn, Ann. Global Anal. Geom., 2015, 47(2), 179-222

  • [12] Kolar I., Michor P. W., Slovak J., Natural operations in differential geometry, Springer, 1993

  • [13] Souriau J.-M., Un algorithme générateur de structures quantiques, Astérisque (hors série), 1985, 341-399

  • [14] Iglesias-Zemmour P., Diffeology, Mathematical Surveys and Monographs, 2013, 185

  • [15] Neeb K.-H., Towards a Lie theory of locally convex groups, Japanese J. Math., 2006, 1, 291-468

  • [16] Christensen D., Sinnamon G., Wu E., The D-topology for diffeological spaces, Pacific J. Math., 2014, 272(1), 87-110

  • [17] Magnot J.-P., q-deformed Lax equations and their differential geometric background, Lambert Academic Publishing, Saarbrucken, Germany, 2015

  • [18] Dugmore D., Ntumba P., On tangent cones of Frölicher spaces, Quaetiones Mathematicae, 2007, 30(1), 67-83

  • [19] Christensen D., Wu E., Tangent spaces and tangent bundles for diffeological spaces, Cahiers de Topologie et Géométrie Différentielle, 2016, LVII, 3-50

  • [20] Leslie J., On a diffeological group realization of certain generalized symmetrizable Kac-Moody Lie algebras, J. Lie Theory, 2003, 13, 427-442

  • [21] Berger M., A panoramic overview of Riemannian geometry, Springer, 2003

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