projective structure and holonomy

Question

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???

Answer 1

There are two kinds of holonomy, which are well contrasted with each other in the opening paragraphs of the link given.

The first kind is exemplified by the holonomy of a Riemannian metric: parallel transport of tangent vectors around a closed loop. This kind of holonomy does not need to be locally trivial, in fact the real usefulness of this holonomy is that it can be used to quantify infinitesmal invariants of Riemannian metrics such as the curvature, by considering the holonomy around very tiny closed loops.

The other kind is exemplified by the holonomy of a flat connection or a projective structure. In contexts like that the holonomy IS required to be locally trivial, meaning that if the closed loop is contained in an open disc then whatever object is being parallel transported (not usually just a tangent vector, perhaps some more complicated object like some kind of transformation such as a projective map), the parallel transport around the loop is the identity. It follows that holonomy around any homotopically trivial closed loop is the identity. It follows that there is a "holonomy homomorphism" defined on the fundamental group. Sometimes the term monodromy is used in this context instead of holonomy. The reader has to understand the ambiguities of the terminology and which meaning is intended by the writer.

In the context of Choi's paper, a projective structure is defined, at the beginning of the article, by local coordinate charts with locally trivial holonomy. He builds up the projective structure first on the complement of the one-skeleton. Then by proving that the holonomy is trivial around each edge, he extends the projective structure to the complement of the zero-skeleton. Then by proving that holonomy is trivial around each vertex, he extends the projective structure to the whole manifold.