Pairs of foliations on surfaces

Author(s)

Regilene Delazari dos Santos Oliveira

• Document type: Doctoral Thesis
• Press: São Carlos.
• Institution: Universidade de São Paulo (USP). Instituto de Ciências Matemáticas e de Computação (ICMC/SB)
• Defense date: 2000-05-24

Examining board members

Farid Tari; Maria Aparecida Soares Ruas; Angela Maria Sitta; Marco Antonio Teixeira; Carlos Teobaldo Gutierrez Vidalon

• Advisor: Farid Tari

Abstract

In this thesis we study in a systematic way pairs of germs of foliations defined by differential 1-forms in the plane using three different techniques: singularity theory, polar blowing up and formal reduction. Using singularity theory, we present a smooth and complete classification of pairs of germs of finite codimension of regular and regular/singular exact foliations in the plane and in 2- dimensional manifolds with regular boundary. We investigate the geometry of the classified pairs and associate invariants to them. With the blowing up method we obtain topological models of pairs of singular and regular/ singular foliations in the plane, when the singularity is of type saddle, node or focus. We also study theses cases using the formal reduction technique. Finally, we prove a desingularization theorem for pairs of germs of differential 1-forms in the plane, similar to Seidenberg and Dumortier\'s theorems for 1-form. (AU)