From Riemannian geometries on spaces of plane curves to vanishing geodesic distance on spaces of submanifolds and of diffeomorphisms.

Joint work with David Mumford.

We study some Riemannian metrics on the space of regular smooth curves in the plane, viewed as the orbit space of immersions from $S1$ to the plane modulo the group of diffeomorphisms of $S1$, acting as reparameterizations. In particular we investigate the metric for a constant $A> 0$: \begin{displaymath} G^A_c(h,k) := \int_{S1}(1+A\kappa_c(\theta)2)\langle h(\theta),k(\theta)\rangle |c'(\theta)|\,d\theta \end{displaymath} where $\kappa_c$ is the curvature of the curve $c$ and $h,k$ are normal vector fields to $c$. For $A=0$, the geodesic distance between any two distinct curves is 0, while for $A>0$ the distance is always positive. We give some lower bounds for the distance function, derive the geodesic equation and the sectional curvature, solve the geodesic equation with simple endpoints numerically, and pose some open questions. The space has an interesting split personality:
among large smooth curves, all its sectional curvatures are $\ge 0$, while for curves with high curvature or perturbations of high frequency, the curvatures are $\le 0$.

In fact, many results hold in a much more general situation. The $L2$-metric or Fubini-Study metric on the non-linear Grassmannian of all submanifolds of type $M$ in a Riemannian manifold $(N,g)$ induces geodesic distance 0. We discuss another metric which involves the mean curvature and shows that its geodesic distance is a good topological metric. The vanishing phenomenon for the geodesic distance holds also for all diffeomorphism groups for the $L2$-metric. This is in particular true for Burgers' equation.