Introduction to the Geometry of Foliations, Part a: Foliations on Compact Surfaces, Fundamentals for Arbitrary Codimension, and Holonomy

ISBN-13: 9783528185015 / Angielski / Miękka / 1986 / 236 str.

Gilbert Hector; Ulrich Hirsch

Opis

Foliation theory grew out of the theory of dynamical systems on manifolds and Ch. Ehresmann's connection theory on fibre bundles. Pioneer work was done between 1880 and 1940 by H. Poincare, I. Bendixson, H. Kneser, H. Whitney, and IV. Kaplan - to name a few - who all studied "regular curve families" on surfaces, and later by Ch. Ehresmann, G. Reeb, A. Haefliger and otners between 1940 and 1960. Since then the subject has developed from a collection of a few papers to a wide field of research. ~owadays, one usually distinguishes between two main branches of foliation theory, the so-called quantitative theory (including homotopy theory and cnaracteristic classes) on the one hand, and the qualitative or geometrie theory on the other. The present volume is the first part of a monograph on geometrie aspects of foliations. Our intention here is to present some fundamental concepts and results as weIl as a great number of ideas and examples of various types. The selection of material from only one branch of the theory is conditioned not only by the authors' personal interest but also by the wish to give a systematic and detailed treatment, including complete proofs of all main results. We hope that tilis goal has been achieved.

Spis treści

Content.- Chatter I — Foliations on Compact Surfaces.- 1. Vector fields on surfaces.- 1.1. Examples of isolated singularities.- 1.2. The indes of an isolated singularity.- 1.3. The theorem of Poincaré — Bohl — Hopf.- 1.4. Existence of non-singular vector fields.- 2. Foliation on surfaces.- 2.1. Motivating remarks.- 2.2. Definition of foliations and related notions.- 2.3. Orientability; relation with vector fields.- 2.4. The existence theorem of Poincaré-Kneser.- 3. Construction of foliations.- 3.1. Suspension.- 3.2. Germs near circle leaves; leaf holonomy.- 3.3. Reeb components.- 3.4. Turbulization.- 3.5. Gluing foliations together.- 4. Classification of foliations on surfaces.- 4.1. Topological dynamics.- 4.2. Foliations on the annulus and on the Möbius band.- 4.3. Foliations on the torus and on the Klein bottle.- 5. Denjoy theory on the circle.- 5.1. The rotation number.- 5.2. Denjoy’s example.- 5.3. Denjoy’s theorem.- 6. Structural stability.- 6.1. Structural stability for diffeomorphisms of the interval and the circle.- 6.2. Structural stability for suspensions.- 6.3. Structural stability for foliations in general.- II — Fundamentals on Foliations.- 1. Foliated bundles.- 1.1. Preparatory material on fibre bundles.- 1.2. Suspensions of group actions.- 1.3. Foliated bundles.- 1.4. Equivariant submersions.- 2. Foliated manifolds.- 2.1. Definition of a foliation; related notions.- 2.2. Transversality; orientability.- 2.3. The tangent bundle of a foliation; Frobenius’ theorem.- 2.4. Pfaffian forms; Frobenius’ theorem (dual version).- 3. Examples of foliated manlfolds.- 3.1. Follations defined by locally free group actions.- 3.2. Foliations with a transverse structure.- III — Holonomy.- 1. Foliated micro bundles.- 1.1. Localization in follated bundles.- 1.2. Generalities on follated microbundles.- 1.3. Holonomy of foliated microbundles.- 2. Holonomy of leaves.- 2.1. Unwrapping of leaves; leaf holonomy.- 2.2. Holonomy and foliated cocycles; leaves without holonomy.- 3. Linear holonomy; Thurston’s stability theorem.- 3.1. Linear and infinitesimal holonomy.- 3.2. Thurston’s stability theorem.- Literature.- Glossary of notations.

O autorze

Prof. Dr. Ulrich Hirsch ist geschäftsführender Gesellschafter der Ulrich Hirsch & Partner Unternehmensberater in Bonn.