William Goldman used projective invariants in order to classify a triangle with three adjacent ones . Choi in " Geometric structures on low-dimensional manifolds " wants to generalize Goldman's results to 3-dimensional manifolds , so he also used the projective invariants so that he could classify a tetrahedron with four adjacent ones. He also introduced some concepts such as " vertex diagram " and " geometric vertex diagram ". In theorem 3 of the above article , he wants to reduce the problem of the existence of real projective structure on a topologically triangulated 3-manifold to one on projective spheres. In his proof of theorem 3 in his article , he said that : Given a tehtrahedron and four adjacent ones in the triangulation of M , we can identify them with ones in the projective space as given by the invariants.such an indentification may not be unique but is unique up to isotopies preserving the triangulatiosn of M. this gives an atlas of charts from M removed with one-skeletons to RP3.since one can send a quintuple of points in general position in RP3 to an arbitrary quintuple by a unique projective map, we see that once we determine the image of the 5-tuples of tetrahedrons , their projective transition maps are determined. if we only consider the interiors of the tetrahedra , we obtain a projective structure on M removed with 1-skeletons.one needs to verify that around the edges , the identifications give us trivial holonomy elemenst .
Why one has to show that around the edges , the identifications give us trivial holonomy elements???