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Master's Dissertation
DOI
10.11606/D.45.2017.tde-17082017-225043
Document
Author
Full name
Bruno de Lessa Victor
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2017
Supervisor
Committee
Cordaro, Paulo Domingos (President)
Kirilov, Alexandre
Silva, Paulo Leandro Dattori da
Title in Portuguese
Resolubilidade semiglobal e global para uma classe de campos vetoriais complexos em variedades diferenciáveis
Keywords in Portuguese
Folheação
Resolubilidade global
Resolubilidade semiglobal
Abstract in Portuguese
Neste trabalho estudamos a resolubilidade suave de campos vetoriais complexos suaves da forma L = L1 + iL2, em uma variedade M, com as seguintes propriedades: em cada ponto de M, os campos L1 e L2 são linearmente independentes , e seu colchete [L1, L2](x) é uma combinação linear de L1(x) e L2(x). Para tratar da resolubilidade local, nos utilizamos da teoria dos espaços Bp,k e operadores de força constante. Seguindo para a resolubilidade semiglobal, estudamos a folheação gerada por L1 e L2: mostramos que neste caso as folhas possuem estrutura de variedade complexa, o que nos permite obter um panorama bastante completo sobre o problema. Para encerrar, provamos que L é globalmente resolúvel se e somente se for semiglobalmente resolúvel e M for L-convexa; exibimos condições suficientes para que isto ocorra.
Title in English
Semi-global and global solvability for a class of complex vector fields in differentiable manifolds
Keywords in English
Foliation
Global solvability
Semi-global solvability
Abstract in English
In this work we shall study the smooth solvability of smooth complex vector fields L = L1 + iL2 on a smooth manifold M, assuming the following properties: for any point of M, L1 and L2 are linearly independent and [L1,L2] is a linear combination of L1 and L2. Discussing local solvability, we shall employ the theory of Bp,k Spaces and Operators of Constant Strength. Moving on to Semi-Global Solvability, we shall study the foliation that is generated by L1 and L2: we prove that in this case the leaves are actually complex manifolds, which allow us to obtain an wide comprehension of the problem. Finally, we show that L is globally solvable if and only if it is semi-globally solvable and M is L-convex; we then exhibit sufficient conditions in order to it occur.
 
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Publishing Date
2017-08-18
 
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