*Non-singularity of bending lengths near convex structures.*

**Abstract:**
Suppose that N is a geometrically finite hyperbolic 3-manifold with
boundary, such that the bending locus of the boundary of its convex core
is a collection of simple closed curves a_1,..., a_n.
We give topological conditions on the a_i for such a convex
structure to exist, and prove that the map which associates to each
structure the hyperbolic lengths of the curves a_i is a global
diffeomorphism from
the set of all convex structures bent along a_1,..., a_n
onto its image. If the set of curves is maximal, their traces (or
complex lengths) are local parameters for the representation space
R(N).