Binary differential equations and differential geometry

Author(s)

Ana Claudia Nabarro

• Document type: Master's Dissertation
• Press: São Carlos.
• Institution: Universidade de São Paulo (USP). Instituto de Ciências Matemáticas e de Computação (ICMC/SB)
• Defense date: 1997-03-24

Examining board members

Maria Aparecida Soares Ruas; Ronaldo Alves Garcia; Farid Tari

• Advisor: Maria Aparecida Soares Ruas

Abstract

A binary differential equation is an implicit differential equation of the form a(x, y)dy2 + 2b(x,y)dxdy + c(x , y)dx2 = O, where a, b, c are smooth functions of x and y. At a point (x, y) where the discriminant, Δ(x, y) =b2 (x, y) a(x,y)c(x,y), is greather or equal than zero, the equation defines a pair of directions in the plane. A natural way to study this equation is to lift these bivalued direction fields to a single field defined on a covering space associated to the set Δ = {(x, y)/b2(x, y) a(x,y)c(x,y) > 0}. A. Davydov [Dv], following the pioneer work of L. Dara classified generic bivalued fields when the set Δ is a smooth curve. J. W. Bruce e F. Tari estudied in [BT a topological classification of the integral curves of the equation when the function Δ(x, y) presents singularity of Morse type. Their approac,h is to reduce the implicit equation to a normal form. The purpose of this work is to study the binary differential equations, in the neighbourhood of one isolated singular point. An analysis of these singularities is made through informations given by the Taylor\'s ,polynomial of the functions a, b e c, without reducing the EDB to a normal form. The results are applied to the study of the lines of curvature of surfaces in R3 and to the study of the asymptotic lines of convex embeddings of surfaces em R4. (AU)