Riemannian exponential maps of the diffeomorphism group of T2

Asian Journal of Mathematics

Volumn 12, Issue 3, 2008, Pages 391-420

  Kappeler, Thomas ^a   Loubet, Enrique ^a   Topalov, Peter ^b  

^a UNIVERSITY OF ZURICH   (Switzerland)

^b NORTHEASTERN UNIVERSITY   (United States)

Author keywords

Diffeomorphism group of the torus; Geodesic exponential maps

Indexed keywords

EID: 58049093395     PISSN: 10936106     EISSN: None     Source Type: Book Series    
DOI: 10.4310/AJM.2008.v12.n3.a7     Document Type: Article

References (29)

1. V. ARNOLD, Sur la géometrie differentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluids parfaits, Ann. Inst. Fourier, 16, (1966), pp. 319-361.
2. V. ARNOLD AND B. KHESIN, Topological methods in hydrodynamics, Springer, 1998.
3. J. BURGERS, A mathematical model illustrating the theory of turbulence, Adv. Appl. Mech., 1 (1948), pp. 171-199.
4. R. CAMASSA AND D. HOLM, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett, 71 (1993), pp. 1661-1664.
5. A. CONSTANTIN AND J. ESCHER, Well-posedness, global existence, and blow-up phenomena for a periodic quasi-linear hyperbolic equation, Commun. Pure Appl. Math., 51 (1998), pp. 475-504.
6. A. CONSTANTIN, T. KAPPELER, B. KOLEV AND P. TOPALOV, On geodesic exponential maps of the Virasoro group, Ann. Glob. Anal. Geom., 31 (2007), pp. 155-180.
7. A. CONSTANTIN AND B, KOLEV, On the geometric approach to the motion of inertial mechanical systems, J. Phys. A, 35 (2002), pp. 1151-1179.
8. A. CONSTANTIN AND B. KOLEV, Geodesic flow on the diffeomorphism group of the circle, Comment. Math. Helv., 78 (2003), pp. 787-804.
9. A. CONSTANTIN AND H. MCKEAN, A shallow water equation on the circle, Commun. Pure Appl. Math., 52 (1999), pp. 949-982.
10. C. DE LELLIS, T. KAPPELER AND P. TOPALOV, Low-regularity solutions of the periodic Camassa-Holm, equation, Comm. in PDE, 32 (2007), pp. 87-126.
11. D. EBIN, The manifold of riemannian metrics, in Global Analysis, Proc. Sympos. Pure Math., vol 15, Amer. Math. Soc., eds. S.S. Chern and S. Smale, 11-40, and Bull. Amer. Math. Soc., 74 (1968), pp. 1001-1003.
12. D. EBIN AND J. MARSDEN, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. Math., 92 (1970), pp. 102-163.
13. A. FoKAS AND B. FUCHSSTEINER, Symplectic structures, their Bäcklund transformation and hereditary symmetries, Physics, D, 4 (1981), pp. 47-66.
14. R. HAMILTON, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc., 7 (1982), pp. 66-222.
15. D. HOLM, J. MARSDEN, T. RATIU AND S. SHKOLLER, The Euler-Poincaré equations and semi-direct products with applications to continuum theories, Adv. Math., 137 (1998), pp. 1-81.
16. T. KAPPELER, E. LOUBET AND P. TOPALOV, Analyticity of Riemannian exponential maps on Diff (T), J. of Lie Theory 17 (2007), pp. 481-503.
17. T. KAPPELER AND J. PÖSCHEL, KdV&KAM, Springer-Verlag, Berlin, 2003.
18. T, KATO, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Ration. Mech. Anal., 58 (1975), pp. 181-205.
19. B. KHESIN AND G. MISIOLEK, Euler equations on homogeneous spaces and Virasoro orbits, Adv. Math, 176 (2003), pp. 116-144.
20. S. KOURANBAEVA, The Camassa-Holm, equation as a geodesic flow on the diffeomorphism group, J. Math. Phys., 40 (1999), pp. 857-868.
21. A. KRIEGL AND P. MICHOR, The Convenient Setting of Global Analysis, AMS Series: Mathematical surveys and monographs, Vol. 53, 1997.
22. S. LANG, Differential manifolds, Addison-Wesley Series in Mathematics, 1972.
23. P. LAX, Functional Analysis, Wiley, 2002.
24. J. MILNOR, Remarks on infinite-dimensional Lie groups, Les Houches, Session XL, 1983, Elsevier Science Publishers B.V., 1984.
25. G. MISIOLEK, A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys., 24 (1998), pp. 203-208.
26. G. MISIOLEK, Classical solutions of the periodic Camassa-Holm equation, GAFA, 12 (2002), pp. 1080-1104.
27. H. OMORI, On the group of diffeomorphisms on a compact manifold, in Global Analysis, Proc. Sympos. Pure Math., 15, eds. S.S. Chern and S. Smale, Amer. Math. Soc., pp. 167-183.
28. R. PALAIS, Foundations of global non-linear analysis, Benjamin, N.Y. 1968.
29. R. PALAIS, Seminar on the Atiyah-Singer index theorem, Princeton, 1965.