Geometria extrínseca de campos de vetores em R3

Abstract

In this work we first consider regular vector fields : R3 􀀀! R3 and its orthogonal distribution of planes. We present a characterization of the normal curvature associated to and the system of implicit differential equations 2(D (dr); dr; ) + h rot( ); i hdr; dri = 0; hdr; i = 0; which define two one-dimensional singular and orthogonal foliations, which we call by principal foliations and whose leaves are the principal lines of the distribution . Next we describe the configurations of the principal foliations in a neighborhood of the generic singular points that constitutes a regular curve in R3, which are denoted by Darbouxian umbilic partially points and semi-Darbouxian. We proceed by studying the stability of the closed principal lines and we also present a Kupka- Smale genericity result. To conclude, we study the structure of the singularities of the principal foliations in a neighborhood of a singular hyperbolic point of the vector field .