Geometric structures on low-dimensional manifolds

Abstract

Geometric structures on 3-manifolds are often projectivelyflat structures. Projectively flat structures on 3-manifolds are givenby atlases of charts to bR P^3 with projective transition maps.Equivalently, they are given by projectivelyflat torsion-free connections. We study the question ofputting projective structures on 3-manifolds.This is done by triangulating a given 3-manifold,and then reducing the question to a 2-dimensionalclassical projective geometry problem produced bythe Haken diagram of the 3-manifold. Next, we show thatthe 2-dimensional problem can be reduced to solvinga system of homogeneous equations that are in product forms ofscalar triple products of vectors. Finally, we will computethe deformation spaces of projective structures ona small class of 3-orbifolds.