Abstract
Foliation Theory is the qualitative study of differential equations. The chapter presents definitions, some examples, and the fundamental concepts like holonomy and transverse structures of foliations. Some themes in the point of view of Differential Geometry are discussed: characteristic classes, basic Hodge theory, and deformations. All foliations considered are regular, that is, all leaves have the same dimension. The theory of singular foliations and specially holomorphic singular foliations is well developed with a plentiful literature. Codimension one foliations constitute a rich theme, which was studied extensively by many people. The richness comes from the existence, for such foliations, of nonsingular transverse vector fields, which give a way to go from a leaf to another. Most of the results in Foliation Theory were first obtained in the codimension one case; the chapter summarizes some of those results.
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