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Some Topics on Geometric Theory of Foliations, Bruno Azevedo Scárdua
The Geometric Theory of Foliations of is one of the fields in Mathematics that joins several distinct domains: Topology, Dynamical Systems, Differential Topology and Geometry, among others. Its great development has allowed a better comprehension of several phenomena of mathematical and physical nature.
Theorems, nowadays considered to be classical, like the Reeb Stability Theorem, Haefliger's Theorem, and Novikov's Compact leaf Theorem, are now searched for holomorphic foliations. Several authors have began to investigate such phenomena (e.g. C. Camacho, A. Lins Neto, E. Ghys, M. Brunella, R. Moussu, S. Novikov and others). The study of such field presumes a knowledge of results and techniques from the real case, and nice familiarity with the classical aspects of Holomorphic Dynamical Systems.
These notes are merely introductory and cover only a minor part of the basic aspects of the rich theory of foliations. In particular, we do not mention the celebrated Theorem of Stability of Reeb.
Rigorous proofs and extensive information may be searched in the bibliography we give at the end of the text. Nevertheless, we have tried to shed some light on the geometry of some classical results and to provide motivation for further study. Our goal is to highlight this geometrical viewpoint despite the loss of some formalism. We hope that this text may be useful to those who appreciate Mathematics, and specially to students that may be interested in this beautiful and fruitful field of Mathematics. The notion of foliation has been originally conceived in a classical approach by C. Ehresmann and G. Reeb by the year of 1950 (see [2],[15], [16]. This concept can be motivated by several different situations in Mathematics. In this part of the text we shall give some examples of such situations.