The geometry of whips

TL;DR

This paper explores the geometric structure of the space of unit-speed arcs, modeling whip motion, revealing complex curvature properties, and comparing it to shape recognition metrics, with implications for understanding geodesic behavior.

Contribution

It establishes the submanifold structure of the arc space, analyzes the exponential map's regularity, and compares the geometric properties to those of the Michor-Mumford shape metric.

Findings

The space of arcs is a submanifold with a non-smooth projection.

The exponential map is continuous and differentiable but not C^1.

The curvature is positive but unbounded, leading to conjugate points at short times.

Abstract

In this paper we study geometric aspects of the space of arcs parametrized by unit speed in the $L^2$ 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 $\eta_{tt} = \partial_s(\sigma \eta_s)$, with $\lvert \eta_s\rvert\equiv 1$ and $\sigma$ given by $\sigma_{ss}- \lvert \eta_{ss}\rvert^2 \sigma = -\lvert \eta_{st}\rvert^2$, with boundary conditions $\sigma(t,1)=\sigma(t,-1)=0$ and $\eta(t,0)=0$. 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 $C^1$. 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 $L^2$ metric introduced by Michor and Mumford for shape recognition on the homogeneous space $\text{Imm}(I, \mathbb{R}^2)/\mathcal{D}(I)$ 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 $L^2$ metric on the arc space yields a genuine Riemannian distance.