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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. (English) Zbl 0953.53002
Progress in Mathematics (Boston, Mass.). 152. Boston, MA: Birkhäuser. xix, 585 p. \$ 89.95 (1999).

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 $\lambda \ge 0·$ 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 ${\pi }_{1}\left(M\right),$ 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 numbers is discussed; “Pinching and Collapse”, where collapse (that is some deviation of the local geometry of $M$ from the Euclidean) for general Riemannian manifolds $M$ with bounded sectional curvature is studied), eleven new sections (one (Degrees of short maps between compact and noncompact manifolds) in Chapter 2, two (First-order metric invariants and ultralimits; Convergence with control) in Chapter 3, two (Unstable systolic inequalities and filling; Finer inequalities and systoles of universal spaces) in Chapter 4, five (Applications of the packing inequalities; On the nilpotency of ${\pi }_{1};$ Simplicial volume and entropy; Generalized simplicial norms and the metrization of homotopy theory; Ricci curvature beyond coverings) in Chapter 5 and one (The Varopoulos isoperimetric inequality) in Chapter 6) and three new Appendices (“Metric Spaces and Mappings Seen at Many Scales”, written by S. Semmes, containing several key ideas of real analysis for spaces and functions which have bounds on their snapshots, made accessible to geometers, and dealing with the interplay between measure theory and the geometry of snapshots; “Paul Levy’s Isoperimetric Inequality”, reproducing the author’s 1980-rendition of this inequality [see the author, Publ. Math. de l’IHES, 1980)]; “Systolically Free Manifolds”, where M. Katz gives an overview of systolic freedom).

This book will become one of the standard references in the research literature on the subject. Many fascinating open problems are pointed out. Since this domain has dramatically exploded since 1979 and also is one which has many contact points with diverse areas of mathematics, it is no small task to present a treatment which is at once broad and coherent. It is a major accomplishment of Misha Gromov to have written this exposition. It is hard work to go through the book, but it is worth the effort.

##### MSC:
 53-02 Research monographs (differential geometry) 53C70 Direct methods ($G$-spaces of Busemann, etc.) 53C20 Global Riemannian geometry, including pinching 57N65 Algebraic topology of manifolds 51K99 Distance geometry