Evolution of curvature invariants and lifting integrability

Abstract

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 yield 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.