H^o type Riemannian metrics on the space of planar curves

Abstract

Michor and Mumford have shown that the distances between planar curves in the simplest metric (not involving derivatives) are identically zero. We consider two conformally equivalent metrics for which the distances between curves are nontrivial. We show that in the case of the simpler of the two metrics, the only minimal geodesics are those corresponding to curve evolution in which the points of the curve move with the same normal speed. An equation for the geodesics and a formula for the sectional curvature are derived; a necessary and sufficient condition for the sectional curvature to be bounded is given.