Geometric structures on low-dimensional manifolds

Abstract

Geometric structures on 3-manifolds are often projectively flat structures. Projectively flat structures on 3-manifolds are given by atlases of charts to RP3 with projective transition maps. Equivalently, they are given by projectively flat torsion-free connections. We study the question of putting projective structures on 3-manifolds. This is done by triangulating a given 3 manifold, and then reducing the question to a 2-dimensional classical projective geometry problem produced by the Haken diagram of the 3-manifold. Next, we show that the 2-dimensional problem can be reduced to solving a system of homogeneous equations that are in product forms of scalar triple products of vectors. Finally, we will compute the deformation spaces of projective structures on a small class of 3-orbifolds.