| Dewey Class |
516.36 |
| Title |
Differential Geometry of Foliations ([EBook] :) : The Fundamental Integrability Problem / / by Bruce L. Reinhart. |
| Author |
Reinhart, Bruce L. |
| Other name(s) |
SpringerLink (Online service) |
| Publication |
Berlin, Heidelberg : : Springer Berlin Heidelberg, , 1983. |
| Physical Details |
X, 196 p. : online resource. |
| Series |
Ergebnisse der Mathematik und ihrer Grenzgebiete, A Series of Modern Surveys in Mathematics 0071-1136 ; ; 99 |
| ISBN |
9783642690150 |
| Summary Note |
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.: |
| Contents note |
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. |
| System details note |
Online access to this digital book is restricted to subscription institutions through IP address (only for SISSA internal users) |
| Internet Site |
http://dx.doi.org/10.1007/978-3-642-69015-0 |
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