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Completely integrable curve flows on adjoint orbits. (English) Zbl 1023.37041

This paper gives a systematic approach of the construction of integrable geometric curve flows on adjoint orbits from flows in the soliton hierarchy associated to a compact Lie group. Here the main tool is the development map which is a bijection between a space of smooth curves on the adjoint orbit and a space of curves in the Lie algebra associated with soliton equations, by which Hamiltonian structures, Bäcklund transformations and finite type solutions of these geometric curve flows are constructed. Several explicit examples of these constructions are given as well.

MSC:

37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K25 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry
53C35 Differential geometry of symmetric spaces
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