Skip to Main Content

Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Conjugate points in the Bott-Virasoro group and the KdV equation
HTML articles powered by AMS MathViewer

by Gerard Misiolek PDF
Proc. Amer. Math. Soc. 125 (1997), 935-940 Request permission


We study the geometry of a right invariant metric on a central extension $\widehat {\mathcal {D}}(S^1)$ of the diffeomorphism group of a circle (the Bott-Virasoro group) introduced by Ovsienko and Khesin. We obtain an expression for the curvature tensor of this metric and apply it to find conjugate points in $\widehat {\mathcal {D}}(S^1)$.
  • V. Arnold, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits, Ann. Inst. Fourier (Grenoble) 16 (1966), no. fasc. 1, 319–361 (French). MR 202082, DOI 10.5802/aif.233
  • A. Bloch, P.S. Krishnaprasad, J.E. Marsden and T. Ratiu, The Euler-Poincare equations and double bracket dissipation, Comm. Math. Phys. 175 (1996), 1–42.
  • J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations, Geom. Funct. Anal. 3 (1993), no. 2, 107–156. MR 1209299, DOI 10.1007/BF01896020
  • Raoul Bott, On the characteristic classes of groups of diffeomorphisms, Enseign. Math. (2) 23 (1977), no. 3-4, 209–220. MR 488080
  • Jeff Cheeger and David G. Ebin, Comparison theorems in Riemannian geometry, North-Holland Mathematical Library, Vol. 9, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. MR 0458335
  • David G. Ebin and Jerrold Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math. (2) 92 (1970), 102–163. MR 271984, DOI 10.2307/1970699
  • I. M. Gel′fand and D. B. Fuks, Cohomologies of the Lie algebra of vector fields on the circle, Funkcional. Anal. i Priložen. 2 (1968), no. 4, 92–93 (Russian). MR 0245035
  • A. A. Kirillov, The orbits of the group of diffeomorphisms of the circle, and local Lie superalgebras, Funktsional. Anal. i Prilozhen. 15 (1981), no. 2, 75–76 (Russian). MR 617476
  • Peter D. Lax, Periodic solutions of the KdV equation, Comm. Pure Appl. Math. 28 (1975), 141–188. MR 369963, DOI 10.1002/cpa.3160280105
  • Jerrold E. Marsden and Tudor S. Ratiu, Introduction to mechanics and symmetry, Texts in Applied Mathematics, vol. 17, Springer-Verlag, New York, 1994. A basic exposition of classical mechanical systems. MR 1304682, DOI 10.1007/978-1-4612-2682-6
  • Gerard Misiołek, Stability of flows of ideal fluids and the geometry of the group of diffeomorphisms, Indiana Univ. Math. J. 42 (1993), no. 1, 215–235. MR 1218713, DOI 10.1512/iumj.1993.42.42011
  • V. Yu. Ovsienko and B. A. Khesin, The super Korteweg-de Vries equation as an Euler equation, Funktsional. Anal. i Prilozhen. 21 (1987), no. 4, 81–82 (Russian). MR 925082
  • Tudor Raţiu, On the smoothness of the time $t$-map of the KdV equation and the bifurcation of the eigenvalues of Hill’s operator, Global analysis (Proc. Biennial Sem. Canad. Math. Congr., Univ. Calgary, Calgary, Alta., 1978) Lecture Notes in Math., vol. 755, Springer, Berlin, 1979, pp. 248–294. MR 564906
  • Graeme Segal, Unitary representations of some infinite-dimensional groups, Comm. Math. Phys. 80 (1981), no. 3, 301–342. MR 626704, DOI 10.1007/BF01208274
  • Graeme Segal, The geometry of the KdV equation, Internat. J. Modern Phys. A 6 (1991), no. 16, 2859–2869. Topological methods in quantum field theory (Trieste, 1990). MR 1117753, DOI 10.1142/S0217751X91001416
Similar Articles
  • Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 58D05, 35Q53
  • Retrieve articles in all journals with MSC (1991): 58D05, 35Q53
Additional Information
  • Gerard Misiolek
  • Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
  • Email:
  • Received by editor(s): October 11, 1995
  • Communicated by: Hal L. Smith
  • © Copyright 1997 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 125 (1997), 935-940
  • MSC (1991): Primary 58D05; Secondary 35Q53
  • DOI:
  • MathSciNet review: 1363431