Foliations: geometric studies.
(Feuilletages: études géométriques.)

*(French)*Zbl 0724.58002
Progress in Mathematics, 98. Basel etc.: Birkhäuser Verlag. xiii, 474 p. sFr. 168.00 (1991).

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 cycles. Both the real and the complex cases are involved.

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 cycles. Both the real and the complex cases are involved.

Reviewer: J.Chrastina (Brno)

##### MSC:

58-02 | Research exposition (monographs, survey articles) pertaining to global analysis |

57R30 | Foliations in differential topology; geometric theory |

57-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to manifolds and cell complexes |

58H05 | Pseudogroups and differentiable groupoids |

58-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to global analysis |

01A65 | Development of contemporary mathematics |

37C85 | Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\) |

37C75 | Stability theory for smooth dynamical systems |