Riemannian metrics on the space of curves are used in shape
analysis to describe deformations that take one shape to another and to
define a distance between shapes. The space of curves is an infinite
dimensional manifold and the Riemannian metrics that are of interest in
shape analysis are weak, i.e., interpreted as maps from the tangent bundle
to its dual, they are injective, but not surjective. One of the
consequences is that the geodesic distance induced by these metrics can
vanish. This talk will present examples, where this happens, and then show
how to modify the metric to prevent the vanishing of the distance. The
second part of the talk will focus on a particular class of metrics,
metrics of Sobolev type, and discuss their mathematical properties.