# Differential Geometry of Foliations

## The Fundamental Integrability Problem

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

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

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.

Blätterung (Math.) Differentialgeometrie Geometry Riemannian manifold diffeomorphism differential geometry manifold

- DOI https://doi.org/10.1007/978-3-642-69015-0
- Copyright Information Springer-Verlag Berlin Heidelberg 1983
- Publisher Name Springer, Berlin, Heidelberg
- eBook Packages Springer Book Archive
- Print ISBN 978-3-642-69017-4
- Online ISBN 978-3-642-69015-0
- Series Print ISSN 0071-1136
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