Moving frames, Geometric Poisson brackets and the KdV-Schwarzian evolution of pure spinors  [ Les repères mobiles, les crochets de Poisson géométriques et l’évolution de KdV-Schwarz pour les spineurs purs ]
Annales de l'Institut Fourier, Tome 61 (2011) no. 6, p. 2405-2434
Nous décrivons un repère mobile non local pour les courbes de spineurs purs dans O(2m,2m)/P, et la base correspondante d’invariants différentiels. Nous montrons que l’espace des invariants différentiels de type Schwarzien définit une sous-variété de crochets de Poisson géométriques de spineurs purs. La restriction résultante est donnée par un systéme découplé de crochets de Poisson de KdV . Nous définissons une généralisation de l’évolution de Schwarz-KdV pour les courbes de spineurs purs et nous montrons que, en restriction à un niveau fixé, cela induit un système d’équations de KdV découplé pour les invariants de type projectif. Nous décrivons par ailleurs la transformation correspondante de Miura et le système non commutatif modifié de KdV.
In this paper we describe a non-local moving frame along a curve of pure spinors in O(2m,2m)/P, and its associated basis of differential invariants. We show that the space of differential invariants of Schwarzian-type define a Poisson submanifold of the spinor Geometric Poisson brackets. The resulting restriction is given by a decoupled system of KdV Poisson structures. We define a generalization of the Schwarzian-KdV evolution for pure spinor curves and we prove that it induces a decoupled system of KdV equations on the invariants of projective type, when restricted to a certain level set. We also describe its associated Miura transformation and non-commutative modified KdV system.
DOI : https://doi.org/10.5802/aif.2678
Classification:  37K,  53D55
Mots clés: repères mobiles, evolution de spineurs, crochet de Poisson géométriques, équations de KdV, invariants différentiels, transformation de Miura, système non commutatif modifié de KdV
@article{AIF_2011__61_6_2405_0,
     author = {Mar\'\i\ Beffa, Gloria},
     title = {Moving frames, Geometric Poisson brackets and the KdV-Schwarzian evolution of pure spinors},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {61},
     number = {6},
     year = {2011},
     pages = {2405-2434},
     doi = {10.5802/aif.2678},
     mrnumber = {2976316},
     zbl = {1245.53066},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2011__61_6_2405_0}
}
Marí Beffa, Gloria. Moving frames, Geometric Poisson brackets and the KdV-Schwarzian evolution of pure spinors. Annales de l'Institut Fourier, Tome 61 (2011) no. 6, pp. 2405-2434. doi : 10.5802/aif.2678. http://www.numdam.org/item/AIF_2011__61_6_2405_0/

[1] Anco, Stephen C. Bi-Hamiltonian operators, integrable flows of curves using moving frames and geometric map equations, J. Phys. A, Tome 39 (2006) no. 9, pp. 2043-2072 | Article | MR 2211976 | Zbl 1085.37049

[2] Anco, Stephen C. Hamiltonian flows of curves in G/ SO (N) and vector soliton equations of mKdV and sine-Gordon type, SIGMA Symmetry Integrability Geom. Methods Appl., Tome 2 (2006), pp. Paper 044, 17 pp. (electronic) | Article | MR 2217753 | Zbl 1102.37042

[3] Bailey, Toby N.; Eastwood, Michael G. Conformal circles and parametrizations of curves in conformal manifolds, Proc. Amer. Math. Soc., Tome 108 (1990) no. 1, pp. 215-221 | Article | Zbl 0684.53016

[4] Bailey, Toby N.; Eastwood, Michael G. Complex paraconformal manifolds—their differential geometry and twistor theory, Forum Math., Tome 3 (1991) no. 1, pp. 61-103 | Article | MR 1085595 | Zbl 0728.53005

[5] Calini, Annalisa; Ivey, Thomas; Marí-Beffa, Gloria Remarks on KdV-type flows on star-shaped curves, Phys. D, Tome 238 (2009) no. 8, pp. 788-797 | Article | MR 2522973 | Zbl 1218.37102

[6] Cartan, Élie La Méthode du Repère Mobile, la Théorie des Groupes Continus et les Espaces Généralisés, Exposés de Géométrie, Hermann, Paris, Tome 5 (1935) | Zbl 0010.39501

[7] Cartan, Élie Les espaces à connexion conforme, Oeuvres Complètes, Gauthier-Villars, Paris, Tome III.1 (1955), pp. 747-797

[8] Chou, Kai-Seng; Qu, Changzheng Integrable equations arising from motions of plane curves, Phys. D, Tome 162 (2002) no. 1-2, pp. 9-33 | Article | MR 1882237 | Zbl 0987.35139

[9] Chou, Kai-Seng; Qu, Changzheng Integrable equations arising from motions of plane curves. II, J. Nonlinear Sci., Tome 13 (2003) no. 5, pp. 487-517 | Article | MR 1882237 | Zbl 1045.35063

[10] Drinfeld, V. G.; Sokolov, V. V. Lie algebras and equations of Korteweg-de Vries type, Current problems in mathematics, Vol. 24, Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow (Itogi Nauki i Tekhniki) (1984), pp. 81-180 | MR 760998 | Zbl 0558.58027

[11] Dubrovin, B. A.; Novikov, S. P. Hydrodynamics of weakly deformed soliton lattices. Differential geometry and Hamiltonian theory, Uspekhi Mat. Nauk, Tome 44 (1989) no. 6(270), p. 29-98, 203 | Article | MR 1037010 | Zbl 0712.58032

[12] Eastwood, M.; Marí Beffa, G. Geometric Poisson brackets on Grassmannian and conformal spheres (submitted)

[13] Fels, Mark; Olver, Peter J. Moving coframes. I. A practical algorithm, Acta Appl. Math., Tome 51 (1998) no. 2, pp. 161-213 | Article | MR 1620769 | Zbl 0937.53012

[14] Fels, Mark; Olver, Peter J. Moving coframes. II. Regularization and theoretical foundations, Acta Appl. Math., Tome 55 (1999) no. 2, pp. 127-208 | Article | MR 1681815 | Zbl 0937.53013

[15] Fialkov, A. The Conformal Theory of Curves, Transactions of the AMS, Tome 51 (1942), pp. 435-568 | MR 6465 | Zbl 0063.01358

[16] Gay-Balmaz, François; Ratiu, Tudor S. Group actions on chains of Banach manifolds and applications to fluid dynamics, Ann. Global Anal. Geom., Tome 31 (2007) no. 3, pp. 287-328 | Article | MR 2314804 | Zbl 1122.58006

[17] Green, Mark L. The moving frame, differential invariants and rigidity theorems for curves in homogeneous spaces, Duke Math. J., Tome 45 (1978) no. 4, pp. 735-779 http://projecteuclid.org/getRecord?id=euclid.dmj/1077313097 | Article | MR 518104 | Zbl 0414.53039

[18] Griffiths, P. On Cartan’s method of Lie groups and moving frames as applied to uniqueness and existence questions in differential geometry, Duke Math. J., Tome 41 (1974), pp. 775-814 | Article | MR 410607 | Zbl 0294.53034

[19] Hasimoto, R. A soliton on a vortex filament, J. Fluid Mechanics, Tome 51 (1972), pp. 477-485 | Article | Zbl 0237.76010

[20] Hitchin, N. J.; Segal, G. B.; Ward, R. S. Integrable systems, The Clarendon Press Oxford University Press, New York, Oxford Graduate Texts in Mathematics, Tome 4 (1999) (Twistors, loop groups, and Riemann surfaces, Lectures from the Instructional Conference held at the University of Oxford, Oxford, September 1997) | MR 1723384 | Zbl 1082.37501

[21] Huang, Rongpei; Singer, David A. A new flow on starlike curves in 3 , Proc. Amer. Math. Soc., Tome 130 (2002) no. 9, p. 2725-2735 (electronic) | Article | MR 1900890 | Zbl 1007.53007

[22] Hubert, E. Generation properties of differential invariants in the moving frame methods (preprint http://hal.inria.fr/inria-00194528/en)

[23] Kirillov, A. A. 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 | Zbl 0427.22012

[24] Langer, Joel; Perline, Ron Poisson geometry of the filament equation, J. Nonlinear Sci., Tome 1 (1991) no. 1, pp. 71-93 | Article | MR 1102831 | Zbl 0795.35115

[25] Langer, Joel; Perline, Ron Geometric realizations of Fordy-Kulish nonlinear Schrödinger systems, Pacific J. Math., Tome 195 (2000) no. 1, pp. 157-178 | Article | MR 1781618 | Zbl 1115.37353

[26] Mansfield, Elizabeth L.; Van Der Kamp, Peter H. Evolution of curvature invariants and lifting integrability, J. Geom. Phys., Tome 56 (2006) no. 8, pp. 1294-1325 | Article | MR 2236264 | Zbl 1099.53012

[27] Marí Beffa, G.; Olver, P. J. Poisson structures for geometric curve flows in semi-simple homogeneous spaces, Regul. Chaotic Dyn., Tome 15 (2010) no. 4-5, pp. 532-550 | Article | MR 2679763 | Zbl pre05836818

[28] Marí Beffa, G.; Sanders, J. A.; Wang, Jing Ping Integrable systems in three-dimensional Riemannian geometry, J. Nonlinear Sci., Tome 12 (2002) no. 2, pp. 143-167 | Article | MR 1894465 | Zbl 1140.37361

[29] Marí Beffa, Gloria On completely integrable geometric evolutions of curves of Lagrangian planes, Proc. Roy. Soc. Edinburgh Sect. A, Tome 137 (2007) no. 1, pp. 111-131 | Article | MR 2359775 | Zbl 1130.37032

[30] Marí Beffa, Gloria Geometric Poisson brackets in flat semisimple homogenous spaces, Asian Journal of Mathematics, Tome 12 (2008) no. 1, pp. 1-33 | MR 2415008 | Zbl 1173.37054

[31] Marí Beffa, Gloria Geometric realizations of bi-Hamiltonian completely integrable systems, SIGMA Symmetry Integrability Geom. Methods Appl., Tome 4 (2008), pp. Paper 034, 23 | Article | MR 2393293 | Zbl 1157.37336

[32] Marí Beffa, Gloria Projective-type differential invariants and geometric curve evolutions of KdV-type in flat homogeneous manifolds, Ann. Inst. Fourier (Grenoble), Tome 58 (2008) no. 4, pp. 1295-1335 http://aif.cedram.org/item?id=AIF_2008__58_4_1295_0 | Article | Numdam | MR 2427961 | Zbl 1192.37099

[33] Marí Beffa, Gloria On bi-Hamiltonian flows and their realizations as curves in real homogeneous manifold, Pacific Journal of Mathematics, Tome 247 (2010) no. 1, pp. 163-188 | Article | MR 2718210 | Zbl 1213.37096

[34] Olver, Peter J. Equivalence, invariants, and symmetry, Cambridge University Press, Cambridge (1995) | Article | MR 1337276 | Zbl 0837.58001

[35] Ovsienko, Valentin Lagrange Schwarzian derivative and symplectic Sturm theory, Ann. Fac. Sci. Toulouse Math. (6), Tome 2 (1993) no. 1, pp. 73-96 | Article | Numdam | MR 1230706 | Zbl 0780.34004

[36] Ovsienko, Valentin; Tabachnikov, S. Projective differential geometry old and new, Cambridge University Press, Cambridge, Cambridge Tracts in Mathematics, Tome 165 (2005) (From the Schwarzian derivative to the cohomology of diffeomorphism groups) | MR 2177471 | Zbl 1073.53001

[37] Sanders, Jan A.; Wang, Jing Ping Integrable systems in n-dimensional Riemannian geometry, Mosc. Math. J., Tome 3 (2003) no. 4, pp. 1369-1393 | MR 2058803 | Zbl 1050.37035

[38] Segal, Graeme The geometry of the KdV equation, Internat. J. Modern Phys. A, Tome 6 (1991) no. 16, pp. 2859-2869 (Topological methods in quantum field theory (Trieste, 1990)) | Article | MR 1117753 | Zbl 0741.35073

[39] Squires, Shane A.; Beffa, Gloria Marí Integrable systems associated to curves in flat Galilean and Minkowski spaces, Appl. Anal., Tome 89 (2010) no. 4, pp. 571-592 | Article | MR 2647767 | Zbl 1218.37095

[40] Terng, Chuu-Lian; Thorbergsson, Gudlaugur Completely integrable curve flows on adjoint orbits, Results Math., Tome 40 (2001) no. 1-4, pp. 286-309 (Dedicated to Shiing-Shen Chern on his 90th birthday) | MR 1860376 | Zbl 1023.37041

[41] Terng, Chuu-Lian; Uhlenbeck, Karen Schrödinger flows on Grassmannians, Integrable systems, geometry, and topology, Amer. Math. Soc., Providence, RI (AMS/IP Stud. Adv. Math.) Tome 36 (2006), pp. 235-256 | MR 2222517 | Zbl 1110.37056