The geometry of whips

Abstract

In this article, we study geometric aspects of the space of arcs parameterized 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(σ ηs), with \({\lvert \eta_{s}\rvert\equiv 1}\) and σ given by \({\sigma_{ss}- \lvert \eta_{ss}\rvert^2 \sigma = -\lvert \eta_{st}\rvert^2}\), 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 is 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.