  In this article we investigate a first order reparametrization-invariant
Sobolev metric on the space of immersed curves. Motivated by applications in
shape analysis where discretizations of this infinite-dimensional space are
needed, we extend this metric to the space of Lipschitz curves, establish the
wellposedness of the geodesic equation thereon, and show that the space of
piecewise linear curves is a totally geodesic submanifold. Thus, piecewise
linear curves are natural finite elements for the discretization of the
geodesic equation. Interestingly, geodesics in this space can be seen as
soliton solutions of the geodesic equation, which were not known to exist for
reparametrization-invariant Sobolev metrics on spaces of curves.
