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Metric Foliations and Curvature / by Detlef Gromoll, Gerard Walschap; edited by H. Bass, J. Oesterlé, A. Weinstein
Contributor(s): Resource type: Ressourcentyp: Buch (Online)Book (Online)Language: English Series: Progress in Mathematics ; 268 | SpringerLink BücherPublisher: Basel ; Boston, Mass. ; Berlin : Birkhäuser, 2009Publisher: [Heidelberg] : [Springer], 2009Description: Online-Ressource (digital)ISBN:- 9783764387150
- 128206925X
- 9781282069251
- 516.362
- 516.36
- 516.373 22
- 510
- QA641-670
- QA613.62
Contents:
Summary: Submersions, Foliations, and Metrics -- Basic Constructions and Examples -- Open Manifolds of Nonnegative Curvature -- Metric Foliations in Space Forms.Summary: In the past three or four decades, there has been increasing realization that metric foliations play a key role in understanding the structure of Riemannian manifolds, particularly those with positive or nonnegative sectional curvature. In fact, all known such spaces are constructed from only a representative handful by means of metric fibrations or deformations thereof. This text is an attempt to document some of these constructions, many of which have only appeared in journal form. The emphasis here is less on the fibration itself and more on how to use it to either construct or understand a metric with curvature of fixed sign on a given space.PPN: PPN: 1647816041Package identifier: Produktsigel: ZDB-2-SEB | ZDB-2-SXMS | ZDB-2-SMA
CONTENTS; Preface; 1 Submersions, Foliations, and Metrics; 1.1 Notation and basic geometric concepts; 1.2 Metric foliations and Riemannian submersions; 1.3 Horizontal lifts and transversal holonomy; 1.4 The fundamental tensors of a submersion; 1.5 Curvature relations; 1.6 Projectable Jacobi fields; 1.7 The Riccati equation for Jacobi fields; 1.8 The dual foliation; 1.9 Basic identities; 2 Basic Constructions and Examples; 2.1 General vertical warping; 2.2 Warped products; 2.3 Homogeneous submersions; 2.4 Left-invariant metrics on Lie groups; 2.5 The Aloff-Wallach examples
2.6 Bi-quotients of Lie groups2.7 Associated bundles; 2.8 Fat bundles; 3 Open Manifolds of Nonegative Curvature; 3.1 Convex sets in Riemannian manifolds; 3.2 The soul construction; 3.3 The topological structure of M; 3.4 The Sharafutdinov retraction; 3.5 The metric projection on to the soul; 3.6 The metric structure of bundles with K = 0; 4 Metric Foliations in Space Forms; 4.1 Isoparametric foliations; 4.2 Metric fibrations of Euclidean space; 4.3 Metric foliations of spheres; 4.4 Geometry of the tangent bundle; 4.5 Compact space forms of nonpositive curvature; Bibliography; Index
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