Metric Foliations and Curvature

By: Detlef Gromoll, Gerard Walschap

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Hardcover | 19 February 2009

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190 Pages

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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.

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From the reviews: "The book under review is one of five or six books on foliations that should be in the professional library of every geometer. ! authors define the fundamental tensors of a Riemannian submersion tensors that carry over to a metric foliation on M ! . gives a brief introduction to the geometry of the second tangent bundle and related topics needed for the study of metric foliations on compact space forms of non negative sectional curvature ! ." (Richard H. Escobales, Jr., Mathematical Reviews, Issue 2010 h)

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Non-Fiction Mathematics Geometry Differential & Riemannian Geometry Topology Analytic Topology

Preface	p. vii
Submersions, Foliations, and Metrics
Notation and basic geometric concepts	p. 1
Metric foliations and Riemannian submersions	p. 5
Horizontal lifts and transversal holonomy	p. 10
The fundamental tensors of a submersion	p. 14
Curvature relations	p. 24
Projectable Jacobi fields	p. 32
The Riccati equation for Jacobi fields	p. 35
The dual foliation	p. 40
Basic identities	p. 44
Basic Constructions and Examples
General vertical warping	p. 45
The connection	p. 45
The curvature tensor	p. 47
The sectional curvatures	p. 52
The Ricci curvature	p. 53
Warped products	p. 56
Homogeneous submersions	p. 63
Left-invariant metrics on Lie groups	p. 72
The Aloff-Wallach examples	p. 82
Bi-quotients of Lie groups	p. 87
The Gromoll-Meyer exotic sphere	p. 88
The seven-dimensional Eschenburg examples	p. 90
Associated bundles	p. 92
Fat bundles	p. 103
Open Manifolds of Nonnegative Curvature
Convex sets in Riemannian manifolds	p. 109
The soul construction	p. 113
The topological structure of M	p. 120
The Sharafutdinov retraction	p. 122
The metric projection onto the soul	p. 126
The metric structure of bundles with K >q; 0	p. 130
Metric Foliations in Space Forms
Isoparametric foliations	p. 135
Metric fibrations of Euclidean space	p. 140
Metric foliations of spheres	p. 149
Geometry of the tangent bundle	p. 157
Compact space forms of nonpositive curvature	p. 161
Bibliography	p. 165
Index	p. 173
Table of Contents provided by Ingram. All Rights Reserved.

Metric Foliations and Curvature

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