Poisson geometry of differential invariants of curves in some nonsemisimple homogeneous spaces

Beffa, G.

doi:10.1090/S0002-9939-05-07998-0

Poisson geometry of differential invariants of curves in some nonsemisimple homogeneous spaces
HTML articles powered by AMS MathViewer

by G. Marí Beffa PDF

Proc. Amer. Math. Soc. 134 (2006), 779-791 Request permission

Abstract:

In this paper we describe a family of compatible Poisson structures defined on the space of coframes (or differential invariants) of curves in flat homogeneous spaces of the form $\mathcal {M} \cong (G\ltimes \mathbb {R}^n)/G$ where $G\subset {\mathrm {GL}}(n,\mathbb {R})$ is semisimple. This includes Euclidean, affine, special affine, Lorentz, and symplectic geometries. We also give conditions on geometric evolutions of curves in the manifold $\mathcal {M}$ so that the induced evolution on their differential invariants is Hamiltonian with respect to our main Hamiltonian bracket.

References

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
Aaron Fialkow, The conformal theory of curves, Trans. Amer. Math. Soc. 51 (1942), 435–501. MR 6465, DOI 10.1090/S0002-9947-1942-0006465-X
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
R. Hasimoto. A soliton on a vortex filament, J. Fluid Mechanics, 51:477–485, 1972.
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
Kai-Seng Chou and Changzheng Qu, Integrable equations arising from motions of plane curves, Phys. D 162 (2002), no. 1-2, 9–33. MR 1882237, DOI 10.1016/S0167-2789(01)00364-5
K.-S. Chou and C.-Z. Qu, Integrable equations arising from motions of plane curves. II, J. Nonlinear Sci. 13 (2003), no. 5, 487–517. MR 2007575, DOI 10.1007/s00332-003-0570-0
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
Franco Magri, A simple model of the integrable Hamiltonian equation, J. Math. Phys. 19 (1978), no. 5, 1156–1162. MR 488516, DOI 10.1063/1.523777
G. Marí Beffa. Hamiltonian structures on the space of differential invariants of curves in flat semisimple homogeneous manifolds , submitted, 2004.
G. Marí Beffa. Poisson brackets associated to the conformal geometry of curves, To appear in the Transactions of the American Mathematical Society.
G. Marí Beffa, Poisson brackets associated to invariant evolutions of Riemannian curves, Pacific J. Math. 215 (2004), no. 2, 357–380. MR 2068787, DOI 10.2140/pjm.2004.215.357
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, 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
Peter J. Olver, Equivalence, invariants, and symmetry, Cambridge University Press, Cambridge, 1995. MR 1337276, DOI 10.1017/CBO9780511609565
Peter J. Olver, Moving frames and singularities of prolonged group actions, Selecta Math. (N.S.) 6 (2000), no. 1, 41–77. MR 1771216, DOI 10.1007/s000290050002
L. V. Ovsiannikov, Group analysis of differential equations, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1982. Translated from the Russian by Y. Chapovsky; Translation edited by William F. Ames. MR 668703
Andrew Pressley and Graeme Segal, Loop groups, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1986. Oxford Science Publications. MR 900587
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
Jan A. Sanders and Jing Ping Wang, Integrable systems in $n$-dimensional Riemannian geometry, Mosc. Math. J. 3 (2003), no. 4, 1369–1393 (English, with English and Russian summaries). MR 2058803, DOI 10.17323/1609-4514-2003-3-4-1369-1393