Geometry and Topology of Convex Projective Manifolds

Florida State University, Tallahassee FL

Investigators

Abstract

Projective geometry is the geometry of perspective, whose practitioners over time have included Greek philosopher/mathematicians studying incidence properties of lines in the plane, Renaissance artists attempting to render more realistic frescoes, and computer scientists pioneering computer graphics and vision techniques. This geometry comes from projecting points in a higher dimensional space along lines to a lower dimensional projective space. Unlike Euclidean geometry, this geometry has no well-defined notions of distance or angle. Its only meaningful geometric notion is incidence (for example, intersections of lines and inclusion of points in lines). In principle, this inability to measure distance initially seems like a drawback; however, in practice it provides a unified framework for studying seemingly disparate and incongruous geometries simultaneously. For example, projective space has pieces that serve as models for the familiar Euclidean geometry, the non-Euclidean spherical and hyperbolic geometries, and other exotic geometries, such as de Sitter and anti de Sitter space, that are of interest in modern physics. Recently, there has been increased interest in properly convex domains, which are interesting pieces of projective space that share many properties with hyperbolic space but enjoy interesting deformation properties absent in the hyperbolic setting. A primary focus of this project is to produce more of these properly convex examples and to understand their geometric, dynamic, and algebraic properties in a systematic fashion. Due to built-in connections with perspective and computer vision, many of the low dimensional examples under study in this project can be effectively visualized and rendered with the aid of a computer to produce vibrant dynamic graphics. This feature will allow the involvement of students with limited mathematical background in portions of the research as well as conveying the spirit of many of the important results to the broader non-mathematical community. Properly convex domains are subsets of projective space that are disjoint from a projective hyperplane and convex in the affine space produced by removing such a hyperplane from projective space. Hyperbolic space serves as the prime example of a properly convex domain via the Klein model. Properly convex domains and their quotients by discrete groups share many properties with hyperbolic space and hyperbolic orbifolds. A main point of this proposal is to understand how familiar concepts in hyperbolic geometry manifest themselves in properly convex geometry. Three main aims of the project are 1) developing a properly convex theory of Dehn surgery that can be used to produce examples of closed properly convex manifolds from non-compact ones, 2) investigating how dynamic properties of the fundamental group of a projective manifold manifest themselves geometrically, in analogy with geometric finiteness for hyperbolic manifolds, and 3) using properly convex structures to produce subgroups of the special linear group with interesting algebraic properties (such as thinness). In addition to the obvious potential to better understand geometric and dynamical aspects of properly convex manifolds, this project should also yield a deeper understanding of hyperbolic geometry by elucidating which geometric features of hyperbolic geometry are consequences of uniform negative curvature and which are consequences of more general geometric structure.