The deformation theory of hyperbolic cone-3-manifolds with cone-angles less than 2π

We develop the deformation theory of hyperbolic cone-3-manifolds with cone-angles less than 2 pi, that is, contained in the interval (0, 2 pi). In the present paper we focus on deformations keeping the topological type of the cone-manifold fixed. We prove local rigidity for such structures. This gives a positive answer to a question of A Casson.