  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.
