Abstract
In this paper we present an overview of the direct relation between differential invariants of projective type for curves in flat semisimple homegenous spaces and PDEs of KdV type. We describe the progress in the proof of a conjectured Theorem stating that for any such space there are geometric evolutions of curves that induce completely integrable evolutions on their invariants of projective-type. The Theorem also states that these evolutions decouple always into either complexly coupled KdV equations (conformai type), decoupled KdV equations (Lagrangian type) or Adler-Gelfand-Dikii generalized KdV types (projective type). The paper also describes the fundamental role of group-based moving frames in this study.
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References
Stephen Anco, Hamiltonian flows of curves in G/SO(N) and vector soliton equations of mKdV and sine-Gordon type, SIGMA, Vol. 2 (2006).
E. Cartan, La Méthode du Repère Mobile, la Théorie des Groupes Continus, et les Espaces Généralisés, Exposés de Géométrie n. 5, Hermann, Paris, 1935.
E. Cartan, La Théorie des Groupes Finis et Continus et la Géométrie Différentielle Traitées par la Méthode du Repère Mobile, Cahiers Scientifiques, Vol. 18, Gauthier-Villars, Paris, 1937.
M. Fels AND P.J. Olver, Moving coframes. I. A practical algorithm, Acta Appl. Math. (1997), pp. 99–136.
M. Fels AND P.J. Olver, Moving coframes. II. Regularization and theoretical foundations, Acta Appl. Math. (1999), pp. 127–208.
A. Fialkov, The Conformal Theory of Curves, Transactions of the AMS, 51 (1942), pp. 435–568.
P.A. Griffiths, On Cartan’s method of Lie groups and moving frames as applied to uniqueness and existence questions in differential geometry, Duke Math. J., 41 (1974), 775–814.
M.L. Green, The moving frame, differential invariants and rigidity theorems for curves in homogeneous spaces, Duke Math. J., 45 (1978), 735–779.
R. Hasimoto, A soliton on a vortex filament, J. Fluid Mechanics, 51 (1972), 477–485.
S. Kobayashi AND T. Nagano, On filtered Lie Algebras and Geometric Structures I, Journal of Mathematics and Mechanics, 13(5) (1964), 875–907.
J. Langer AND R. Perline, Poisson geometry of the filament equation, J. Nonlinear Sci., 1(1), (1991), 71–93.
G. Marí Beffa, Hamiltonian Structures on the space of differential invariants of curves in flat semisimple homogenous manifolds, submitted.
G. Marí Beffa, Poisson geometry of differential invariants of curves in some nonsemisimple homogenous spaces, Proc. Amer. Math. Soc, 134 (2006), 779–791.
G. Marí Beffa, Projective-type differential invariants and geometric curve evolutions of KdV-type in flat homogeneous manifolds, submitted.
G. Marí Beffa, On completely integrable geometric evolutions of curves of Lagrangian planes, accepted for publication in the Proceedings of the Royal academy of Edinburg, 2006.
G. Marí Beffa, Poisson brackets associated to the conformal geometry of curves, Trans. Amer. Math. Soc, 357 (2005), 2799–2827.
G. Marí Beffa, The theory of differential invariants and KdV Hamiltonian evolutions, Bull. Soc. Math. France, 127(3), (1999), 363–391.
G. Marí Beffa, J. Sanders, AND J.P. Wang, Integrable Systems in Three-Dimensional Riemannian Geometry, J. Nonlinear Sc. (2002), pp. 143–167.
P. Olver, Invariant variational problems and integrable curve flows, presentation, Cocoyoc, Mexico (2005).
T. Ochiai, Geometry associated with semisimple flat homogeneous spaces, Transactions of the AMS, 152 (1970), pp. 159–193.
V. Ovsienko, Lagrange Schwarzian derivative and symplectic Sturm theory, Ann. de la Fac. des Sciences de Toulouse, 6(2), (1993), pp. 73–96.
A. Pressley AND G. Segal, Loop groups, Graduate Texts in Mathematics, Springer, 1997.
J. Sanders AND J.P. Wang, Integrable Systems in n-dimensional Riemannian Geometry, Moscow Mathematical Journal, 3, 2004.
E.J. Wilczynski, Projective differential geometry of curves and ruled surfaces, B.G. Teubner, Leipzig, 1906.
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Beffa, G.M. (2008). Projective-Type Differential Invariants for Curves and Their Associated Pdes of Kdv Type. In: Eastwood, M., Miller, W. (eds) Symmetries and Overdetermined Systems of Partial Differential Equations. The IMA Volumes in Mathematics and its Applications, vol 144. Springer, New York, NY. https://doi.org/10.1007/978-0-387-73831-4_12
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DOI: https://doi.org/10.1007/978-0-387-73831-4_12
Publisher Name: Springer, New York, NY
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