Teorema de Quine para aplicações de Gauss

Abstract

In this thesis we aim to study stable Gauss maps. We ivestigate their singularities, which are organized as simple, closed and disjoint curves, called the singular curves, or, in the case of Gauss maps, parabolic curves. These curves will always be the boundary between a region of positive Gaussian curvature (elliptic region) and a region of negative Gaussian curvature (hyperbolic region). We also study how to associate to each map a bipartite weighted graph, whose vertices are associated to the regular regions, with the sign of each vertex being the sign of the Gaussian curvature of the correspondig region, the edges are associated to the parabolic curves and the weights to the genus of each region. Moreover, the adjacency between vertices reflects the adjacency between the regular regions, making the graph an importan tool for analyzing the topology of the surface. At last, we discuss how Quine’s Theorem appears in the case of stable Gauss maps.