  Pseudospherical surfaces determined by Cauchy problems involving the
Camassa-Holm equation are considered herein. We study how global solutions
influence the corresponding surface, as well as we investigate two sorts of
singularities of the metric: the first one is just when the co-frame of dual
form is not linearly independent. The second sort of singularity is that
arising from solutions blowing up. In particular, it is shown that the metric
blows up if and only if the solution breaks in finite time.
