  We consider a volume maximization program to construct hyperbolic structures
on triangulated 3-manifolds, for which previous progress has lead to consider
angle assignments which do not correspond to a hyperbolic metric on each
simplex. We show that critical points of the generalized volume are associated
to geometric structures modeled on the extended hyperbolic space -- the natural
extension of hyperbolic space by the de Sitter space -- except for the
degenerate case where all simplices are Euclidean in a generalized sense.
  Those extended hyperbolic structures can realize geometrically a
decomposition of the manifold as connected sum, along embedded spheres (or
projective planes) which are totally geodesic, space-like surfaces in the de
Sitter part of the extended hyperbolic structure.
