Conjugate points in the Bott-Virasoro group

and the KdV equation

Author:
Gerard Misiolek

Journal:
Proc. Amer. Math. Soc. **125** (1997), 935-940

MSC (1991):
Primary 58D05; Secondary 35Q53

DOI:
https://doi.org/10.1090/S0002-9939-97-03711-8

MathSciNet review:
1363431

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study the geometry of a right invariant metric on a central extension 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 .

**[A]**V. I. Arnold,*Sur la geometrie differentielle des groupes de Lie de dimension infinie et ses applications a l'hydrodynamique des fluids parfaits*, Ann. Inst. Grenoble**16**(1966). MR**34:1956****[BKMR]**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. CMP**96:07****[B]**J. Bourgain,*Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution eqations, Part II: The KdV equation*, GAFA**3**(1993). MR**95d:35160b****[Bo]**R. Bott,*On the characteristic classes of groups of diffeomorphisms*, Enseign. Math.**23**(1977), 209-220. MR**58:7651****[CE]**J. Cheeger and D.G. Ebin,*Comparison Theorems in Riemannian Geometry*, North Holland, New York 1975. MR**56:16538****[EM]**D.G. Ebin and J. Marsden,*Groups of diffeomorphisms and the motion of an incompressible fluid*, Ann. of Math.**92**(1970), 102-160. MR**42:6865****[GF]**I.M. Gelfand and D.B. Fuchs,*The cohomology of the Lie algebra of vector fields on a circle*, Funkts. Anal. Prilozhen.**2**(1968). MR**39:6348a****[K]**A.A. Kirillov,*Orbits of the group of diffeomorphisms of a circle and local superalgebras*, Funct. Anal. Appl.**15**(1981). MR**82j:17005****[L]**P. Lax,*Periodic solutions of the KdV equations*, Comm. Pure Appl. Math.**28**(1975), 141-188. MR**51:6192****[MR]**J. Marsden and T. Ratiu, Introduction to Mechanics and Symmetry, Springer 1995. MR**95i:58073****[M]**G. Misio{\l}ek,*Stability of flows of ideal fluids and the geometry of the group of diffeomorphisms*, Indiana Univ. Math. J.**42**(1993), 215-235. MR**94j:58027****[OK]**V. Ovsienko and B. Khesin,*Korteweg-de Vries superequations as an Euler equation*, Funct. Anal. Appl.**21**(1987). MR**89g:58095****[R]**T. Ratiu,*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.**755**Springer (1979). MR**81j:58025****[S1]**G. Segal,*Unitary representations of some infinite dimensional groups*, Comm. Math. Phys. 80 (1981), 301-342. MR**82k:22004****[S2]**G. Segal,*The geometry of the KdV equation*, Int. J. Mod. Phys.**6**(1991), 2859-2869. MR**92e:58102**

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:
misiolek.1@nd.edu

DOI:
https://doi.org/10.1090/S0002-9939-97-03711-8

Keywords:
Diffeomorphism groups,
KdV equation,
conjugate points

Received by editor(s):
October 11, 1995

Communicated by:
Hal L. Smith

Article copyright:
© Copyright 1997
American Mathematical Society