Poisson brackets associated to the conformal geometry of curves
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- by G. Marí Beffa PDF
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Abstract:
In this paper we present an invariant moving frame, in the group theoretical sense, along curves in the Möbius sphere. This moving frame will describe the relationship between all conformal differential invariants for curves that appear in the literature. Using this frame we first show that the Kac-Moody Poisson bracket on $Lo(n+1,1)^\ast$ can be Poisson reduced to the space of conformal differential invariants of curves. The resulting bracket will be the conformal analogue of the Adler–Gel’fand–Dikii bracket. Secondly, a conformally invariant flow of curves induces naturally an evolution on the differential invariants of the flow. We give the conditions on the invariant flow ensuring that the induced evolution is Hamiltonian with respect to the reduced Poisson bracket. Because of a certain parallelism with the Euclidean case we study what we call Frenet and natural cases. We comment on the implications for completely integrable systems, and describe conformal analogues of the Hasimoto transformation.References
- Richard L. Bishop, There is more than one way to frame a curve, Amer. Math. Monthly 82 (1975), 246–251. MR 370377, DOI 10.2307/2319846
- Cartan, E. La Méthode du Repère Mobile, la Théorie des Groupes Continus et les Espaces Généralisés”, Exposés de Géométrie 5, Hermann, Paris, 1935.
- G. Darboux. Leçons sur la Géometrie des espaces de Riemann, Gauthier-Villars, 1910.
- A. Doliwa and P. M. Santini, An elementary geometric characterization of the integrable motions of a curve, Phys. Lett. A 185 (1994), no. 4, 373–384. MR 1261407, DOI 10.1016/0375-9601(94)90170-8
- V. G. Drinfel′d and V. V. Sokolov, Lie algebras and equations of Korteweg-de Vries type, Current problems in mathematics, Vol. 24, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1984, pp. 81–180 (Russian). MR 760998
- Sergio Sispanov, Generalización del teorema de Laguerre, Bol. Mat. 12 (1939), 113–117 (Spanish). MR 3
- Mark Fels and Peter J. Olver, Moving coframes. I. A practical algorithm, Acta Appl. Math. 51 (1998), no. 2, 161–213. MR 1620769, DOI 10.1023/A:1005878210297
- Mark Fels and Peter J. Olver, Moving coframes. II. Regularization and theoretical foundations, Acta Appl. Math. 55 (1999), no. 2, 127–208. MR 1681815, DOI 10.1023/A:1006195823000
- Victor Guillemin and Shlomo Sternberg, Variations on a theme by Kepler, American Mathematical Society Colloquium Publications, vol. 42, American Mathematical Society, Providence, RI, 1990. MR 1104658
- R. Hasimoto. A soliton on a vortex filament, J. Fluid Mechanics, 51:477–485, 1972.
- A. A. Kirillov, Infinite-dimensional groups, their representations, orbits, invariants, Proceedings of the International Congress of Mathematicians (Helsinki, 1978) Acad. Sci. Fennica, Helsinki, 1980, pp. 705–708. MR 562675
- Shoshichi Kobayashi, Transformation groups in differential geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 70, Springer-Verlag, New York-Heidelberg, 1972. MR 0355886, DOI 10.1007/978-3-642-61981-6
- Joel Langer and Ron Perline, Poisson geometry of the filament equation, J. Nonlinear Sci. 1 (1991), no. 1, 71–93. MR 1102831, DOI 10.1007/BF01209148
- Joel Langer and Ron Perline, Geometric realizations of Fordy-Kulish nonlinear Schrödinger systems, Pacific J. Math. 195 (2000), no. 1, 157–178. MR 1781618, DOI 10.2140/pjm.2000.195.157
- G. Marí Beffa. Poisson brackets associated to invariant evolutions of Riemannian curves, to appear in the Pacific Journal of Mathematics, 2004.
- Gloria Marí Beffa, The theory of differential invariants and KdV Hamiltonian evolutions, Bull. Soc. Math. France 127 (1999), no. 3, 363–391 (English, with English and French summaries). MR 1724401, DOI 10.24033/bsmf.2353
- G. Marí Beffa. On Relative and Absolute differential invariants of conformal curves, Journal of Lie Theory,13, pp. 213-245, 2003.
- G. Marí Beffa, J. A. Sanders, and Jing Ping Wang, Integrable systems in three-dimensional Riemannian geometry, J. Nonlinear Sci. 12 (2002), no. 2, 143–167. MR 1894465, DOI 10.1007/s00332-001-0472-y
- Jerrold E. Marsden and Tudor Ratiu, Reduction of Poisson manifolds, Lett. Math. Phys. 11 (1986), no. 2, 161–169. MR 836071, DOI 10.1007/BF00398428
- Jerrold Marsden and Alan Weinstein, Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids, Phys. D 7 (1983), no. 1-3, 305–323. Order in chaos (Los Alamos, N.M., 1982). MR 719058, DOI 10.1016/0167-2789(83)90134-3
- Peter J. Olver, Equivalence, invariants, and symmetry, Cambridge University Press, Cambridge, 1995. MR 1337276, DOI 10.1017/CBO9780511609565
- R. W. Sharpe, Differential geometry, Graduate Texts in Mathematics, vol. 166, Springer-Verlag, New York, 1997. Cartan’s generalization of Klein’s Erlangen program; With a foreword by S. S. Chern. MR 1453120
- Chuu-Lian Terng and Gudlaugur Thorbergsson, Completely integrable curve flows on adjoint orbits, Results Math. 40 (2001), no. 1-4, 286–309. Dedicated to Shiing-Shen Chern on his 90th birthday. MR 1860376, DOI 10.1007/BF03322713
- Chuu-Lian Terng and Karen Uhlenbeck, Poisson actions and scattering theory for integrable systems, Surveys in differential geometry: integral systems [integrable systems], Surv. Differ. Geom., vol. 4, Int. Press, Boston, MA, 1998, pp. 315–402. MR 1726931, DOI 10.4310/SDG.1998.v4.n1.a7
- Yukinori Yasui and Norihito Sasaki, Differential geometry of the vortex filament equation, J. Geom. Phys. 28 (1998), no. 1-2, 195–207. MR 1653155, DOI 10.1016/S0393-0440(98)00024-2
Additional Information
- G. Marí Beffa
- Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
- Email: maribeff@math.wisc.edu
- Received by editor(s): June 11, 2003
- Received by editor(s) in revised form: November 14, 2003
- Published electronically: September 23, 2004
- © Copyright 2004 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 357 (2005), 2799-2827
- MSC (2000): Primary 37K25, 37K05, 37K10; Secondary 53A55
- DOI: https://doi.org/10.1090/S0002-9947-04-03589-5
- MathSciNet review: 2139528