My question relates to constructing angled triangulations or hyperbolic triangulations for $3$--manifolds. Briefly, an angle triangulation can be considered as an assignment of a real number (called an angle) to each edge of a tetrahedron such that when we glue the tetrahedron up to obtain the $3$--manifold, around each edge the sum of the angles are $2\pi$, and around each vertex the sum of angles is $\pi$ (we also require that opposite edges of each tetrahedron have the same angle). In short, an angle structure corresponds to the linear part of Thurston's gluing equations -- it corresponds to a weak hyperbolic structure in that the induced metric on the manifold may not be complete. 


Now I have assumed the topological conditions required to obtain a hyperbolic structure on the manifold via Thurston's hyperbolization theorem (i.e., the manifolds I am looking at are compact, irreducible, atoroidal with torus boundary) and I am able to explicitly construct special spines for these. Dual to these special spines are ideal triangulations (this comes from the work of Matveev).


My question is whether there are well known conditions for a special spine to be 'geometric'. That is given a special spine, we can look at the special spine as a special polyhedron onto which the manifold collapses. Does it follow that if I look at the special spine as a hyperbolic polyhedron, and I know that the special spine when thickened gives me a $3$--manifold, does this imply that the $3$--manifold is a hyperbolic $3$--manifold. Any discussion, questions and ideas will be appreciated. This is my first time posting here so apologies in advance if I have not followed protocol.



Standard (special) spines and hyperbolic structure on 3-manifolds
If an ideal triangulation (or its dual spine) admits an angle structure (with *positive* angles), then the manifold must admit a complete hyperbolic metric of finite volume (see [Theorem 10.2](http://www.ams.org/mathscinet-getitem?mr=1985831), an observation of Casson). However, when one straightens these tetrahedra in the complete hyperbolic metric, they might not be positively oriented. Casson also observed (after Igor Rivin) that the volume function on the space of angle structures is maximized at a complete hyperbolic structure; he made some attempt to find a different proof of geometrization of cusped hyperbolic manifolds using this approach (see [Futer-Gueritaud](http://www.ams.org/mathscinet-getitem?mr=2796632) for an exposition). However, as far as I know, there is no nice characterization of when an ideal triangulation with a positive angle structure admits a positive solution to Thurston's gluing equations. 


One slight correction: an angle structure (with positive angles) gives rise to an incomplete hyperbolic metric on the complement of the 1-skeleton. 


