##
**Metric spaces, convexity and nonpositive curvature.**
*(English)*
Zbl 1115.53002

IRMA Lectures in Mathematics and Theoretical Physics 6. Zürich: European Mathematical Society Publishing House (ISBN 3-03719-010-8/pbk). xi, 287 p. (2005).

This book gives the relation between convexity theory and the theory of spaces of nonpositive curvature. First of all, the convexity of the distance function in a non-positively curved space is responsible for many of the global properties of that space. H. Busemann defined nonpositive curvature precisely by a convexity property of the distance function and showed that most of the important properties of a non-positively curved Riemannian manifold are valid in a setting which is much wider than that of Riemannian geometry. Secondly, many of the basic results in convexity theory have the flavor of nonpositive curvature. There is an important class of examples, the local-implies-global properties, such as the fact that a locally convex function is globally convex, or the fact that a local geodesic in a Busemann space, i.e., in a simply connected metric space whose distance function is convex, is a global geodesic. The author describes these facts in detail.

Chapter 1 starts with lengths of paths, differentiable paths in Euclidean spaces, and spaces of paths. Chapter 2 deals with length and geodesic spaces. A length space is a metric space in which the distance between any pair of points is equal to the greatest lower bound of the set of lengths of paths joining them. If the distance between any pair of points is equal to the length of some path joining them, then the space is said to be a geodesic space. A subset \(A\) of a geodesic space \(X\) is geodesically convex if for every two points in \(A\) the geodesic segment connecting them is in \(A\).

In Chapter 3, the author studies properties of maps between metric spaces, mostly distance-preserving surjective maps, or isometries. Besides isometries, there are several other classes of maps between metric spaces: distance-preserving maps, length-preserving maps, which may be considered as the analogs of distance-preserving maps in the setting of length spaces, Lipschitz maps, \(K\)-Lipschitz maps, bi-Lipschitz homeomorphisms, Hölder maps, distance-non-increasing maps, length-non-increasing maps, contractions, distance-decreasing maps, local homeomorphisms, locally \(K\)-Lipschitz maps, local isometries, quasi-conformal maps, quasi-isometries, and covering maps that are local isometries.

In Chapter 4, two generalizations of distances are presented: distances between subsets of metric spaces and between isometries of a metric space. The Hausdorff and the Busemann-Hausdorff distances are the examples of these generalizations.

Chapter 5 introduces convexity in vector spaces including the Minkowski construction and the Hilbert geometry. Convex functions are discussed in Chapter 6. Strictly convex normed vector spaces as examples of geodesic spaces are studied in Chapter 7. In Chapter 8, the author studies the basic properties of Busemann spaces, that is, geodesic metric spaces in which for any two geodesics \(\gamma\) and \(\gamma'\), the distance function \(D(t,t')=d(\gamma(t),\gamma'(t'))\) is convex.

Chapter 9 presents locally convex spaces. A metric space \(X\) is said to be locally convex if every point \(x\in X\) has a neighborhood \(U(x)\) equipped with the induced metric is a Busemann space. The more specialized final Chapters 10, 11, and 12 treat matters such as asymptotic rays, the visual boundary, the classification of isometries in elliptic, parabolic, and hyperbolic cases, Busemann functions, co-rays, and horospheres.

At the end of each chapter there are notes indicating some further developments or presenting some historical aspects. The concepts and techniques used in the book are illustrated by many examples from classical hyperbolic geometry and from the theory of Teichmüller spaces.

This well-written book should be useful for students and researchers in geometry, topology, and analysis.

Chapter 1 starts with lengths of paths, differentiable paths in Euclidean spaces, and spaces of paths. Chapter 2 deals with length and geodesic spaces. A length space is a metric space in which the distance between any pair of points is equal to the greatest lower bound of the set of lengths of paths joining them. If the distance between any pair of points is equal to the length of some path joining them, then the space is said to be a geodesic space. A subset \(A\) of a geodesic space \(X\) is geodesically convex if for every two points in \(A\) the geodesic segment connecting them is in \(A\).

In Chapter 3, the author studies properties of maps between metric spaces, mostly distance-preserving surjective maps, or isometries. Besides isometries, there are several other classes of maps between metric spaces: distance-preserving maps, length-preserving maps, which may be considered as the analogs of distance-preserving maps in the setting of length spaces, Lipschitz maps, \(K\)-Lipschitz maps, bi-Lipschitz homeomorphisms, Hölder maps, distance-non-increasing maps, length-non-increasing maps, contractions, distance-decreasing maps, local homeomorphisms, locally \(K\)-Lipschitz maps, local isometries, quasi-conformal maps, quasi-isometries, and covering maps that are local isometries.

In Chapter 4, two generalizations of distances are presented: distances between subsets of metric spaces and between isometries of a metric space. The Hausdorff and the Busemann-Hausdorff distances are the examples of these generalizations.

Chapter 5 introduces convexity in vector spaces including the Minkowski construction and the Hilbert geometry. Convex functions are discussed in Chapter 6. Strictly convex normed vector spaces as examples of geodesic spaces are studied in Chapter 7. In Chapter 8, the author studies the basic properties of Busemann spaces, that is, geodesic metric spaces in which for any two geodesics \(\gamma\) and \(\gamma'\), the distance function \(D(t,t')=d(\gamma(t),\gamma'(t'))\) is convex.

Chapter 9 presents locally convex spaces. A metric space \(X\) is said to be locally convex if every point \(x\in X\) has a neighborhood \(U(x)\) equipped with the induced metric is a Busemann space. The more specialized final Chapters 10, 11, and 12 treat matters such as asymptotic rays, the visual boundary, the classification of isometries in elliptic, parabolic, and hyperbolic cases, Busemann functions, co-rays, and horospheres.

At the end of each chapter there are notes indicating some further developments or presenting some historical aspects. The concepts and techniques used in the book are illustrated by many examples from classical hyperbolic geometry and from the theory of Teichmüller spaces.

This well-written book should be useful for students and researchers in geometry, topology, and analysis.

Reviewer: Andrew Bucki (Edmond)

### MSC:

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |

53C70 | Direct methods (\(G\)-spaces of Busemann, etc.) |

53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |

53C65 | Integral geometry |

51K05 | General theory of distance geometry |

52A07 | Convex sets in topological vector spaces (aspects of convex geometry) |

52A41 | Convex functions and convex programs in convex geometry |

53C22 | Geodesics in global differential geometry |

54E35 | Metric spaces, metrizability |

57M50 | General geometric structures on low-dimensional manifolds |