An invitation to web geometry. From Abel’s addition theorem to the algebraization of codimension one webs. Paper from the 27th Brazilian Mathematics Colloquium – 27\(^\circ\) Colóquio Brasileiro de Matemática, Rio de Janeiro, Brazil, July 27–31, 2009. (English) Zbl 1184.53002

Publicações Matemáticas do IMPA. Rio de Janeiro: Instituto Nacional de Matemática Pura e Aplicada (IMPA) (ISBN 978-85-2440291-3/pbk). xii, 245 p. (2009).
This is a book written, as a first purpose, to serve as supporting material for a 5-lecture mini course on web geometry (27th Brazilian Mathematical Colloquium). The growth of the material led the authors to a second purpose (which from the point of view of this reviewer is accomplished): provide an account of some of the main results in the field of web geometry of the last years.
The book is structured as follows: In the Introduction some historical notes are provided. Chapter 1 is introductive, basic notions on web geometry are given. Chapter 2 focuses on the notions of abelian relation and rank: Proofs of the Chern’s bound on the rank and of the normal form of the conormals of a web are provided. Abel’s addition Theorem is the goal of Chapter 3, being the converse of Abel’s Theorem proved in Chapter 4. Chapter 5 is devoted to Trépreau’s algebraization result and Chapter 6 takes up the study of planar webs of maximal rank.


53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
53A60 Differential geometry of webs