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On geodesic exponential maps of the Virasoro group

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Abstract

We study the geodesic exponential maps corresponding to Sobolev type right-invariant (weak) Riemannian metrics μ(k) (k≥ 0) on the Virasoro group Vir and show that for k≥ 2, but not for k = 0,1, each of them defines a smooth Fréchet chart of the unital element e ∈Vir. In particular, the geodesic exponential map corresponding to the Korteweg–de Vries (KdV) equation (k = 0) is not a local diffeomorphism near the origin.

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Authors and Affiliations

  1. Department of Mathematics, Lund University, Box 118, SE-22100, Lund, Sweden

    A. Constantin

  2. School of Mathematics, Trinity College Dublin, Dublin 2, Ireland

    A. Constantin

  3. Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057, Zürich, Switzerland

    T. Kappeler

  4. CMI, Université de Provence, 39 rue F. Joliot-Curie, 13453, Marseille Cedex, France

    B. Kolev

  5. Northeastern University, 360 Huntington Avenue, Boston, MA, 02115, USA

    P. Topalov

Authors
  1. A. Constantin
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  2. T. Kappeler
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  3. B. Kolev
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  4. P. Topalov
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Correspondence to T. Kappeler.

Additional information

A. Constantin: Supported in part by the European Community through the FP6 Marie Curie RTN ENIGMA (MRTN-CT-2004-5652).

T. Kappeler: Supported in part by the SNSF, the programme SPECT, and the European Community through the FP6 Marie Curie RTN ENIGMA (MRTN-CT-2004-5652)

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Constantin, A., Kappeler, T., Kolev, B. et al. On geodesic exponential maps of the Virasoro group. Ann Glob Anal Geom 31, 155–180 (2007). https://doi.org/10.1007/s10455-006-9042-8

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