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Evolution of curvature invariants and lifting integrability. (English) Zbl 1099.53012

Summary: Given a geometry defined by the action of a Lie-group on a flat manifold, the Fels-Olver moving frame method yields a complete set of invariants, invariant differential operators, and the differential relations, or syzygies, they satisfy. We give a method that determines, from minimal data, the differential equations the frame must satisfy, in terms of the curvature and evolution invariants that are associated to curves in the given geometry. The syzygy between the curvature and evolution invariants is obtained as a zero curvature relation in the relevant Lie-algebra. An invariant motion of the curve is uniquely associated with a constraint specifying the evolution invariants as a function of the curvature invariants. The zero curvature relation and this constraint together determine the evolution of curvature invariants.
Invariantizing the formal symmetry condition for curve evolutions yields a syzygy between different evolution invariants. We prove that the condition for two curvature evolutions to commute appears as a differential consequence of this syzygy. This implies that integrability of the curvature evolution lifts to integrability of the curve evolution, whenever the kernel of a particular differential operator is empty. We exhibit various examples to illustrate the theorem; the calculations involved in verifying the result are substantial.

MSC:

53A55 Differential invariants (local theory), geometric objects
70G65 Symmetries, Lie group and Lie algebra methods for problems in mechanics
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
14H50 Plane and space curves

Software:

Indiff
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Full Text: DOI

References:

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