Simmetries in binary differential equations

• Main Author: Tempesta, Patricia
• Publication Date: 2017
• Format: Doctoral thesis
• Language: eng
• Source: Biblioteca Digital de Teses e Dissertações da USP
• Download full: http://www.teses.usp.br/teses/disponiveis/55/55135/tde-11072017-170308/
• topic: 1-forma quadratica equivariante; Binary differential equation; Compact Lie group; Equação diferencial binária; Equivariant quadratic 1-form; Grupo de Lie compacto; Representation theory; Simetria; Symmetry; Teoria de representação

The purpose of this thesis in to introduce the systematic study of symmetries in binary differential equations (BDEs). We formalize the concept of a symmetric BDE, under the linear action of a compact Lie group. One of the main results establishes a formula that relates the algebraic and geometric effects of the occurrence of the symmetry in the problem. Using tools from invariant theory and representation theory for compact Lie groups we deduce the general forms of equivariant binary differential equations under compact subgroups of O(2). A study about the behavior of the invariant straight lines on the configuration of homogeneous BDEs of degree n is done with emphasis on cases in which n = 0 and n = 1. Also for the linear case (n = 1) the equivariant normal forms are presented. Symmetries of linear 1-forms are also studied and related with symmetries of tangent orthogonal vectors fields associated with it.