| Differential Geometry of Foliations: The Fundamental Integrability Problem Softcover Repri Edition Contributor(s): Reinhart, B. L. (Author) |
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ISBN: 3642690173 ISBN-13: 9783642690174 Publisher: Springer OUR PRICE: $52.24 Product Type: Paperback - Other Formats Published: January 2012 |
| Additional Information |
| BISAC Categories: - Mathematics | Geometry - Differential |
| Dewey: 516.36 |
| Series: Ergebnisse Der Mathematik Und Ihrer Grenzgebiete. 2. Folge |
| Physical Information: 0.45" H x 6.69" W x 9.61" (0.76 lbs) 196 pages |
| Descriptions, Reviews, Etc. |
| Publisher Description: 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. |