Differential Geometry of Foliations: The Fundamental Integrability Problem

ISBN-13: 9783642690174 / Angielski / Miękka / 2012 / 196 str.

B.L. Reinhart

Opis

Whoever you are How can I but offer you divine leaves . . . ? Walt Whitman The object of study in modern differential geometry is a manifold with a differ ential structure, and usually some additional structure as well. Thus, one is given a topological space M and a family of homeomorphisms, called coordinate sys tems, between open subsets of the space and open subsets of a real vector space V. It is supposed that where two domains overlap, the images are related by a diffeomorphism, called a coordinate transformation, between open subsets of V. M has associated with it a tangent bundle, which is a vector bundle with fiber V and group the general linear group GL(V). The additional structures that occur include Riemannian metrics, connections, complex structures, foliations, and many more. Frequently there is associated to the structure a reduction of the group of the tangent bundle to some subgroup G of GL(V). It is particularly pleasant if one can choose the coordinate systems so that the Jacobian matrices of the coordinate transformations belong to G. A reduction to G is called a G-structure, which is called integrable (or flat) if the condition on the Jacobians is satisfied. The strength of the integrability hypothesis is well-illustrated by the case of the orthogonal group On. An On-structure is given by the choice of a Riemannian metric, and therefore exists on every smooth manifold."

Spis treści

I. Differential Geometric Structures and Integrability.- 1. Pseudogroups and Groupoids.- 2. Foliations.- 3. The Integrability Problem.- 4. Vector Fields and Pfaffian Systems.- 5. Leaves and Holonomy.- 6. Examples of Foliations.- II. Prolongations, Connections, and Characteristic Classes.- 1. Truncated Polynomial Groups and Algebras.- 2. Prolongation of a Manifold.- 3. Higher Order Structures.- 4. Connections and Characteristic Classes.- 5. Foliations, Connections, and Secondary Classes.- III. Singular Foliations.- 1. The Classifying Space for a Topological Groupoid.- 2. Vector Fields and the Cohomology of Lie Algebras.- 3. Frobenius Structures.- IV. Metric and Measure Theoretic Properties of Foliations.- 1. Analytic Background.- 2. Measure, Volume, and Foliations.- 3. Foliations of a Riemannian Manifold.- 4. Riemannian Foliations.- 5. Foliations with a Few Derivatives.- Supplementary Bibliography.- Index of Terminology.- Index of Symbols.