Foliations: geometric studies
type: Item
instance of: scholarly article
title: Foliations: geometric studies
zbMATH Open document ID: 0724.58002
author: C. Godbillon
published in: Progress in Mathematics
publication date: 1992-09-18T00:00:00Z
review text: A foliation of a manifold M is a decomposition of M into disjoint subsets \(L_ a\) (the leaves, \(a\in A)\) such that each point of M has a neighbourhood U with local coordinates \(x^ 1,...,x^ p,y^ 1,...,y^ q\) on U \((p+q=\dim M)\) such that the connected components of every intersection \(L_ a\cap U\) are determined by the equations \(y^ i\equiv c^ i\) \((i=1,...,q\); \(c^ i\) are constants). Owing to Frobenius theorem, the theory of foliations may be regarded as a global geometry of completely integrable Pfaffian systems, in particular, it generalizes the theory of dynamical systems on manifolds. The monograph may be included among the most successful mathematical works. It may be used both as a rather complete survey of an unusual number of diverse results (together with competent historical comments and extensive chronological bibliography (1950-1989) occupying 75 pages) and as an excellent but advanced textbook (since it includes thorough discussion of fundamental concepts, numerous examples, careful proofs of all general results together with an overlook of several auxiliary tools: Lie groups, connections, pseudogroups, growth theory of algebraic systems, etc.). On the other hand, some lengthy and special constructions (e.g., the universal objects, characteristic classes) are either omitted or presented without much details. Moreover, a preliminary acquaintance with rather advanced parts of classical analysis and algebraic topology is assummed. Concerning the contents, the first chapter includes various famous examples and constructions of foliations, the second chapter the holonomy pseudogroups and stability theorems, the third chapter deals with a dozen of transversal structures, the fourth chapter is devoted to codimension one foliations, and the concluding fifth chapter discusses the concept of domination (growth theory) on leaves: the asymptotics of Riemann volumes, of number of elements of the holonomy group, and of transverse measures and asymptotic cy…
Mathematics Subject Classification ID: 58-02
Mathematics Subject Classification ID: 57R30
Mathematics Subject Classification ID: 57-01
Mathematics Subject Classification ID: 58H05
Mathematics Subject Classification ID: 58-01
Mathematics Subject Classification ID: 01A65
Mathematics Subject Classification ID: 37C85
Mathematics Subject Classification ID: 37C75
zbMATH DE Number: 50350
zbMATH Keywords: manifold
zbMATH Keywords: foliations
zbMATH Keywords: survey
zbMATH Keywords: bibliography
zbMATH Keywords: holonomy pseudogroups
zbMATH Keywords: stability theorems
zbMATH Keywords: transversal structures
zbMATH Keywords: codimension one foliations
zbMATH Keywords: domination
zbMATH Keywords: growth
zbMATH Keywords: leaves
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 50350