Congruências de retas e equações das superfícies principais

Abstract

Motivated by Monge, which in [10] presented the principal configuration of tri-axial ellipsoid, and by Sotomayor and Gutierrez, which in [12], using the Qualitative Theory of O.D.E.’s rigorously established the configuration of the lines of curvature in neighborhoods of umbilic points, our objective is, considering the congruence of lines generated by the restriction to the Ellipsoid of a linear field with three real, distinct and nonzero eigenvalues or two complex conjugate eigenvalues and one nonzero real eigenvalue, to study the behavior of the principal curves of the congruence on the Ellipsoid. In this context, Bianchi [1], Eisenhart [6], Forsyth [8], Pottmann and Wallner [17] and Weatherburn [22] are our references for the Theory of Congruences of Lines . The principal curves are integral curves of a Binary Differential Equation, hence, Bruce and Fidal [2] and Bruce and Tari [3] are our references to study the local the behavior of these curves in neighborhoods of special points, that we will call Umbilic Singularities of the Congruence. In the final chapter, what we intend to present as contribution of this study are the possible configurations of the principal curves of the congruence in the Ellipsoid for the considered congruences.