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Differential Geometry of Foliations

The Fundamental Integrability Problem

  • Book
  • © 1983

Overview

Authors:
  1. Bruce L. Reinhart

    1. Department of Mathematics, University of Maryland, College Park, USA

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Part of the book series: Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge (MATHE2, volume 99)

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Table of contents (4 chapters)

  1. Front Matter

    Pages I-X
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  2. Differential Geometric Structures and Integrability

    • Bruce L. Reinhart
    Pages 1-46

  3. Prolongations, Connections, and Characteristic Classes

    • Bruce L. Reinhart
    Pages 47-92

  4. Singular Foliations

    • Bruce L. Reinhart
    Pages 93-107

  5. Metric and Measure Theoretic Properties of Foliations

    • Bruce L. Reinhart
    Pages 108-180

  6. Back Matter

    Pages 181-198
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Keywords

About this book

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.

Authors and Affiliations

  • Department of Mathematics, University of Maryland, College Park, USA

    Bruce L. Reinhart

Bibliographic Information

  • Book Title: Differential Geometry of Foliations

  • Book Subtitle: The Fundamental Integrability Problem

  • Authors: Bruce L. Reinhart

  • Series Title: Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge

  • DOI: https://doi.org/10.1007/978-3-642-69015-0

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer-Verlag Berlin Heidelberg 1983

  • Softcover ISBN: 978-3-642-69017-4Published: 19 January 2012

  • eBook ISBN: 978-3-642-69015-0Published: 06 December 2012

  • Edition Number: 1

  • Number of Pages: X, 196

  • Topics: Differential Geometry

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