  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.
