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On geodesic exponential maps of the Virasoro group. (English) Zbl 1121.35111
Summary: We study the geodesic exponential maps corresponding to Sobolev type right-invariant (weak) Riemannian metrics $\mu^{(k)}$ $(k \geq$ 0) on the Virasoro group $Vir$ and show that for $k \geq 2$, but not for $k = 0,1$, each of them defines a smooth Fréchet chart of the unital element $e \in$ 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.

35Q35PDEs in connection with fluid mechanics
37K30Relations of infinite-dimensional systems with algebraic structures
37K25Relations of infinite-dimensional systems with differential geometry
58B25Group structures and generalizations on infinite-dimensional manifolds
Full Text: DOI
[1] Arnold V. (1966). Sur la geometrié differentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluids parfaits. Ann. Inst. Fourier 16(1):319--361
[2] Arnold V., Khesin B. (1998). Topological Methods in Hydrodynamics. Springer--Verlag, New York · Zbl 0902.76001
[3] Bona, J.-L., Smith, R.: The initial-value problem for Korteweg--de Vries equation. Phil. Trans. R. Soc. Lon. Ser. A, Math. Phys. Sci. 278, 555--601 (1975) · Zbl 0306.35027
[4] Burgers, J.: A mathematical model illustrating the theory of turbulence. Adv. Appl. Mech. 1, 171--199 (1948)
[5] Camassa, R., Holm, D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661--1664 (1993) · Zbl 0972.35521
[6] Constantin, A.: On the Cauchy problem for the periodic Camassa--Holm equation. J. Differ. Eq. 141, 218--235 (1997) · Zbl 0889.35022
[7] Constantin, A., Escher, J.: Well-posedness, global existence, and blow-up phenomena for a periodic quasi-linear hyperbolic equation. Commun. Pure Appl. Math. 51, 475--504 (1998) · Zbl 0934.35153
[8] Constantin, A., Escher, J.: On the blow-up rate and the blow-up set of breaking waves for a shallow water equation. Math. Z. 233, 75--91 (2000) · Zbl 0954.35136
[9] Constantin, A., McKean, H.: A shallow water equation on the circle. Commun. Pure Appl. Math. 52, 949--982 (1999) · Zbl 0940.35177
[10] Constantin, A., Kolev, B.: On the geometric approach to the motion of inertial mechanical systems. J. Phys. A 35, R51--R79 (2002) · Zbl 1039.37068
[11] Constantin, A., Kolev, B.: Geodesic flow on the diffeomorphism group of the circle. Comment. Math. Helv. 78, 787--804 (2003) · Zbl 1037.37032
[12] Constantin, A., Kolev, B., Lenells, J.: Integrability of invariant metrics on the Virasoro group. Phys. Lett. A 350, 75--80 (2006) · Zbl 1195.58006
[13] De Lellis, C., Kappeler, T., Topalov, P.: Low regularity solutions of the Camassa--Holm equation (to appear in Comm. in PDE) · Zbl 1123.35045
[14] Ebin, D., Marsden, J.: Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. Math. 92, 102--163 (1970) · Zbl 0211.57401
[15] Fokas, A., Fuchssteiner, B.: Symplectic structures, their Bäcklund transformation and hereditary symmetries. Physica D 4, 47--66 (1981) · Zbl 1194.37114
[16] Hamilton, R.: The inverse function theorem of Nash and Moser. Bull. Amer. Math. Soc. 7, 66--222 (1982) · Zbl 0499.58003
[17] Holm, D., Marsden, J., Ratiu, T.: The Euler--Poincaré equations and semidirect products with applications to continuum theories. Adv. Math. 137, 1--81 (1998) · Zbl 0951.37020
[18] Kappeler, T., Pöschel, J.: KdV&KAM. Springer-Verlag, Berlin (2003)
[19] Kappeler, T., Topalov, P.: Well-posedness of KdV on H 1 ( $\mathbb{T}$ ) (to appear in Duke Math. J.) · Zbl 1101.35367
[20] Kato, T.: The Cauchy problem for quasi-linear symmetric hyperbolic systems. Arch. Ration. Mech. Anal. 58, 181--205 (1975) · Zbl 0343.35056
[21] Khesin, B., Misiolek, G.: Euler equations on homogeneous spaces and Virasoro orbits. Adv. Math. 176, 116--144 (2003) · Zbl 1017.37039
[22] Kopell, N.: Commuting diffeomorphisms. (Proc. Symp. Pure Math). Am. Math. Soc. 14, 165--184 (1970) · Zbl 0225.57020
[23] Kouranbaeva, S.: The Camassa--Holm equation as a geodesic flow on the diffeomorphism group. J. Math. Phys. 40, 857--868 (1999) · Zbl 0958.37060
[24] Lang, S.: Differential manifolds. Addison-Wesley Series in Mathematics, Mass (1972) · Zbl 0239.58001
[25] Lenells, J.: The correspondence between KdV and Camassa--Holm. Int. Math. Res. Not. 71, 3797--3811 (2004) · Zbl 1082.35134
[26] McKean, H.P.: The Liouville correspondence between the Korteweg--de Vries and the Camassa--Holm hierarchies. Comm. Pure Appl. Math. 56, 998--1015 (2003) · Zbl 1037.37030
[27] Michor, P., Ratiu, T.: On the geometry of the Virasoro--Bott group. J. Lie Theory 8, 293--309 (1998) · Zbl 0945.58005
[28] Milnor, J.: Remarks on infinite-dimensional Lie groups. Les Houches, Session XL, 1983, Elsevier Science Publishers B.V. (1984) · Zbl 0557.10031
[29] Misiolek, G.: A shallow water equation as a geodesic flow on the Bott--Virasoro group. J. Geom. Phys. 24, 203--208 (1998) · Zbl 0901.58022
[30] Misiolek G. (2002). Classical solutions of the periodic Camassa--Holm equation. GAFA 12, 1080--1104 · Zbl 1158.37311 · doi:10.1007/PL00012648
[31] Ovsienko V., Khesin B. (1987). Korteweg--de Vries superequations as an Euler equation. Funct. Anal. Appl. 21, 81--82 · Zbl 0639.58019
[32] Quantum fields and strings: a course for mathematicians. vols. 1 and 2. Deligne, P., Etingof, P., etc., American Mathematical Society, Providence, RI; Institute for Advanced Study, Princeton, NJ (1999)