The geometry of whips

الأصل · EN

In this paper we study geometric aspects of the space of arcs parametrized by unit speed in the L² metric. Physically this corresponds to the motion of a whip, and it also arises in studying shape recognition. The geodesic equation is the nonlinear, nonlocal wave equation ηtt = ∂ₛ(σηₛ), with ηₛ≡ 1 and σ given by σss- ηss² σ= - ηst², with boundary conditions σ(t,1)=σ(t,-1)=0 and η(t,0)=0. We prove that the space of arcs is a submanifold of the space of all curves, that the orthogonal projection exists but is not smooth, and as a consequence we get a Riemannian exponential map that it continuous and even differentiable but not C¹. This is related to the fact that the curvature is positive but unbounded above, so that there are conjugate points at arbitrarily short times along any geodesic. We also compare this metric to an L² metric introduced by Michor and Mumford for shape recognition on the homogeneous space Imm(I, R²)/D(I) of immersed curves modulo reparametrizations; we show it has some similar properties (such as nonnegative but unbounded curvature and a nonsmooth exponential map), but that the L² metric on the arc space yields a genuine Riemannian distance.