Metric Foliations and Curvature

Authors: Detlef Gromoll, Gerard Walschap

Publisher: Birkhäuser Basel

Book Series : Progress in Mathematics

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

About this book

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.

Frontmatter

Chapter 1. Submersions, Foliations, and Metrics

Abstract

The concept of submersion is dual to what is arguably the oldest notion in differential geometry, that of immersion. Both are generalizations of diffeomorphisms. In the presence of a Riemannian metric, it is natural to consider distance-preserving maps rather than diffeomorphisms. These in turn generalize to isometric immersions, and their metric dual, Riemannian submersions.

Chapter 2. Basic Constructions and Examples

Abstract

Any Riemannian submersion can be used to generate new ones by deforming the metric in the vertical direction. To be specific, let π : (M, 〈, 〉 → B be a Riemannian submersion. Given φ : M → ℝ, define a new metric 〈, 〉_φ on M by

$$\left\langle {e,f} \right\rangle _\phi = e^{2\phi (p)} \left\langle {e^v ,f^v } \right\rangle + \left\langle {e^h ,f^h } \right\rangle , e,f \in M_p , p \in M. $$

Since the horizontal metric is unchanged, π : (M, 〈, 〉_φ) → B is still a Riemannian submersion. X, Y, Z will denote basic fields, T_i vertical ones, and $ \tilde \nabla $, $ \tilde R $ the Levi-Civita connection and curvature tensor, respectively, of 〈, 〉_φ. We will assume that the deformation is constant along fibers, or equivalently, that the gradient of φ is basic.

Chapter 3. Open Manifolds of Nonnegative Curvature

Abstract

Noncompact manifolds with a complete metric of nonnegative sectional curvature were studied in detail by Gromoll-Meyer [58], and by Cheeger-Gromoll [36], who gave a thorough account of their topology. Apart from some special cases, however, their metric structure has only been understood fairly recently. It illustrates the key role that Riemannian submersions seem to play in nonnegative curvature.

Chapter 4. Metric Foliations in Space Forms

Abstract

We have so far focused our attention mostly on the base space B of a Riemannian submersion M → B, in particular when searching for new metrics of nonnegative curvature on B. It is also interesting to look at the total space of the fibration. The very fact that there exists a Riemannian submersion from M (or more generally, that M admits a metric foliation) is a sign that the space possesses a fair amount of symmetry. One therefore expects those Riemannian manifolds with the largest amount of symmetry — namely, space forms — to be the ones that display the most variety as far as these foliations are concerned. Surprisingly, a complete classification of metric foliations on spaces of constant curvature is not yet available. There does, however, exist a classification of metric fibrations, at least in nonnegative curvature, which will be described in this chapter.

Backmatter

Title: Metric Foliations and Curvature
Authors: Detlef Gromoll
Gerard Walschap
Publisher: Birkhäuser Basel
Electronic ISBN: 978-3-7643-8715-0
Print ISBN: 978-3-7643-8714-3
DOI: https://doi.org/10.1007/978-3-7643-8715-0

Springer Professional

Metric Foliations and Curvature

About this book

Table of Contents

Frontmatter

Chapter 1. Submersions, Foliations, and Metrics

Chapter 2. Basic Constructions and Examples

Chapter 3. Open Manifolds of Nonnegative Curvature

Chapter 4. Metric Foliations in Space Forms

Backmatter

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