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Comparison theorems in Riemannian geometry. Reprint of the 1975 original. (English) Zbl 1142.53003
Providence, RI: AMS Chelsea Publishing (ISBN 978-0-8218-4417-5/hbk). x, 161 p. (2008).
The central theme of this book is the interaction between the curvature of a complete Riemannian manifold and its topology and global geometry. More precisely, it concerns the following type of question: What are the properties of a manifold \(M\) whose sectional curvature \(K\) verifies certain inequalities?
The exposition of these results occupies the 4 last chapters. We shall first give some indication of their contents.
The first five of the 9 chapters are preparatory in nature. They begin with a very concise introduction to Riemannian geometry, followed by an exposition of Toponogov’s theorem – the first such treatment in a book in English. Next comes a detailed presentation of homogeneous spaces in which the main goal is to find formulas for their curvature. A quick chapter on Morse theory is followed by one on the injectivity radius. Chapters 6–9 deal with many of the most relevant contributions to the subject in the years 1959 to 1974. These include the pinching (or sphere) theorem, Berger’s theorem for symmetric spaces, the differentiable sphere theorem, the structure of complete manifolds of non-negative curvature, and finally, results about the structure of complete manifolds of non-positive curvature. All results of these chapters that we will quote here are fully proven in the book. Chapter 6 is concerned with the pinching (or sphere) theorem and its generalizations, stating that a simply connected manifold whose curvature verifies \(1> K>{1\over 4}\) is homeomorphic to a sphere. If one supposes only \(1\geq K\geq {1\over 4}\) the conclusion fails, but one shows that the only exceptions are symmetric spaces.
A type of property, associated with the replacement of a strict inequality by a weak one, is called “rigidity property” and reappears in other chapters of the book. In chapter 7 (the differentiable sphere theorem), the authors prove the existence of a constant \(\delta<1\), independent of the dimension of \(M\), such that if \(M\) is simply connected and \(1\geq K>\delta\), \(M\) is diffeomorphic to a sphere. Chapter 8 studies the structure of complete manifolds of non-negative curvature. The main result of this chapter is the existence of a soul \(S\) in every non-compact manifold such that \(K\geq 0\), i.e. a compact totally geodesic submanifold which is a totally convex set. The inclusion of \(S\) in \(M\) is a homotopy equivalence, which shows in particular that \(M\) has the homotopy type of a compact manifold. If \(K>0\), \(S\) is reduced to a point and \(M\) is contractible. The existence of \(S\) can be seen as a rigidity property for this last statement.
Chapter 9 deals with the properties of the first homotopy group of compact manifolds of nonpositive curvature. For instance, it is shown that a solvable subgroup of \(\Pi_1(M)\) is Bierbach group associated to a flat submanifold of \(M\). In these chapters, the main tools are the comparison theorems of Rauch and Toponogov. They are proven in chapters 1 and 2. Other ingredients which are used are exposed in the 5 first chapters, making the book accessible to non-specialists. Chapter 1 (Basic concepts and results) deals rapidly with the standard material of differential geometry which is used afterwards. Chapter 3 recalls some results on homogeneous spaces (stating without proofs the underlying results on Lie groups). Chapter 4 deals rapidly with (finite-dimensional) Morse theory and, with more details, with its application on geodesics.
Much of the material, particularly the last four chapters, was essentially state-of-the-art when the book first appeared in 1975 (see Zbl 0309.53035). Since then, the subject has exploded, but the material covered in the book still represents an essential prerequisite for anyone who wants to work in the field. To conclude, one can say that this book presents many interesting and recent results of global Riemannian geometry, and that by its well composed introductory chapters, the authors have managed to make it readable by non-specialists.

53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
53C20 Global Riemannian geometry, including pinching
53C30 Differential geometry of homogeneous manifolds
58E10 Variational problems in applications to the theory of geodesics (problems in one independent variable)
55Q05 Homotopy groups, general; sets of homotopy classes