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Cover: 9783642690174 | Differential Geometry of Foliations | B. L. Reinhart | Taschenbuch
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Differential Geometry of Foliations
The Fundamental Integrability Problem
Taschenbuch von B. L. Reinhart
Sprache: Englisch

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Beschreibung
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.
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.
Inhaltsverzeichnis
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.
Details
Erscheinungsjahr: 2012
Fachbereich: Geometrie
Genre: Mathematik
Rubrik: Naturwissenschaften & Technik
Medium: Taschenbuch
Seiten: 212
Reihe: Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge
Inhalt: x
196 S.
ISBN-13: 9783642690174
ISBN-10: 3642690173
Sprache: Englisch
Ausstattung / Beilage: Paperback
Einband: Kartoniert / Broschiert
Autor: Reinhart, B. L.
Auflage: Softcover reprint of the original 1st ed. 1983
Hersteller: Springer-Verlag GmbH
Springer Berlin Heidelberg
Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge
Maße: 244 x 170 x 12 mm
Von/Mit: B. L. Reinhart
Erscheinungsdatum: 19.01.2012
Gewicht: 0,375 kg
preigu-id: 106332514
Inhaltsverzeichnis
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.
Details
Erscheinungsjahr: 2012
Fachbereich: Geometrie
Genre: Mathematik
Rubrik: Naturwissenschaften & Technik
Medium: Taschenbuch
Seiten: 212
Reihe: Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge
Inhalt: x
196 S.
ISBN-13: 9783642690174
ISBN-10: 3642690173
Sprache: Englisch
Ausstattung / Beilage: Paperback
Einband: Kartoniert / Broschiert
Autor: Reinhart, B. L.
Auflage: Softcover reprint of the original 1st ed. 1983
Hersteller: Springer-Verlag GmbH
Springer Berlin Heidelberg
Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge
Maße: 244 x 170 x 12 mm
Von/Mit: B. L. Reinhart
Erscheinungsdatum: 19.01.2012
Gewicht: 0,375 kg
preigu-id: 106332514
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