Riemannian geometries on spaces of plane curves

Abstract

We study some Riemannian metrics on the space of regular smooth curves in the plane, viewed as the orbit space of maps from $S^1$ to the plane modulo the group of diffeomorphisms of $S^1$, acting as reparameterizations. In particular we investigate the metric for a constant $A> 0$: $$ G^A_c(h,k) := \int_{S^1}(1+A\ka_c(\th)^2)< h(\th),k(\th) > |c'(\th)| d\th $$ where $\ka_c$ is the curvature of the curve $c$ and $h,k$ are normal vector fields to $c$. The term $A\ka^2$ is a sort of geometric Tikhonov regularization because, for A=0, the geodesic distance between any 2 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 $\le 0$.