Metric structures for Riemannian and non-Riemannian spaces. Transl. from the French by Sean Michael Bates. With appendices by M. Katz, P. Pansu, and S. Semmes. Edited by J. LaFontaine and P. Pansu
type: Item
instance of: scholarly article
title: Metric structures for Riemannian and non-Riemannian spaces. Transl. from the French by Sean Michael Bates. With appendices by M. Katz, P. Pansu, and S. Semmes. Edited by J. LaFontaine and P. Pansu
zbMATH Open document ID: 0953.53002
published in: Progress in Mathematics
publication date: 1999-08-05T00:00:00Z
review text: For Riemannian manifolds and mappings between them one can define various metric invariants. The simplest ones are the diameter and the volume of such a manifold, while an important invariant of a mapping between Riemannian manifolds is its dilatation. The classical Gauss-Bonnet theorem gives an upper bound for the diameter of a positively curved manifold, from which one can deduce the finitude of its fundamental group. For a deeper topological study of Riemannian manifolds, one needs more subtle invariants than diameter or volume. Such invariants provide an important link between the given infinitesimal information about the Riemannian manifold (usually expressed as some restriction on the curvature) and the topology of the manifold. This book, representing the English edition of the 1979 French version which appeared in the Éditions Cedic (see Zbl 0509.53034), presents a systematic treatment of these invariants and describes some of the connections between them and the curvature of a Riemannian manifold. Most of these metric invariants transform in a \( \lambda -\)controlled way under \( \lambda -\)Lipschitz maps, which enlarge distances at most by a factor \( \lambda \) for some \( \lambda \geq 0 .\) A special treatment is given to systoles measuring the minimal volumes of homology classes in a Riemannian manifold and to isoperimetric profiles of complete Riemannian manifolds and infinite groups which are linked to quasiconformal and quasiregular mappings. It is explained how the Ricci curvature controls the volumes of balls of a Riemannian manifold \( M ,\) what leads to several topological consequences concerning \( \pi _1 (M), \) and the isoperimetric profile of \( M. \) The book includes many author's contributions in the field. The author has made substantial additions to the French edition: two new chapters (``Convergence and Concentration of Metrics and Measures'', linking geometry and probability theory and where a geometric version of the law of large nu…
reviewed by: Mircea Craioveanu
Mathematics Subject Classification ID: 53-02
Mathematics Subject Classification ID: 53C70
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 57N65
Mathematics Subject Classification ID: 51K99
zbMATH DE Number: 1321104
zbMATH Keywords: Riemannian manifold
zbMATH Keywords: non-Riemannian space
zbMATH Keywords: bounded Ricci curvature
zbMATH Keywords: growth of fundamental groups
zbMATH Keywords: metrization of homotopy theory
zbMATH Keywords: packing inequality
zbMATH Keywords: Loewner's theorem
zbMATH Keywords: Jacobi variety
zbMATH Keywords: systoles
zbMATH Keywords: pinching
zbMATH Keywords: collapse
zbMATH Keywords: isoperimetric inequality
MaRDI profile type: Publication
author: Misha Gromov
description: scientific article; zbMATH DE number 1321104