We propose a generalization of Casson and Rivin’s program of finding
hyperbolic structures on compact ideally triangulated 3-manifolds with
torus boundary to all closed triangulated 3-manifolds. In the
generalization, the concepts of angle structure and its volume are
introduced for 3-simplices and triangulated closed 3-manifolds. It is
shown that the critical points of the volume function defined on the
space of all angle structures are closely related to the constant
curvature metrics on 3-manifolds. It is also shown that the volume
function can be extended continuously to the compact closure of the
space of all angle structures.