Michor and Mumford have shown that the distances between planar curves in the simplest metric (not involving derivatives) are identically zero. We derive geodesic equations and a formula for sectional curvature for conformally equivalent metrics. We show if the conformal factor depends only on the length of the curve, the metric behaves like an L metric, the sectional curvature is not bounded f...

We give a necessary and sufficient condition on the cost function so that the map solution of Monge’s optimal transportation problem is continuous for arbitrary smooth positive data. This condition was first introduced by Ma, Trudinger and Wang [17, 21] for a priori estimates of the corresponding Monge-Ampère equation. It is expressed by a socalled cost-sectional curvature being non-negative. W...