This is a tight, elegant, and delightful addition to the literature on global Riemannian geometry. The authors focus on the geometrical and topological conclusions that can be drawn from assumptions on the curvature. Techniques are developed for comparing the geometry of a general manifold with that of a simply connected model space of constant curvature, e.g., the comparison theorems of Rauch and Toponogov. The tightness of the exposition and a few misprints leave the reader with some challenging work but the essential ideas are always clear. Chapter 1 moves quickly through material on the exponential map, minimizing properties of geodesics, Hopf-Rinow theorem, conjugate points, Jacobi fields, the second variation formula and the index form for fields along a geodesic, the theorems of Myers, Bonnet, Rauch, Cartan-Hadamard, and Cartan-Ambrose-Hicks. Chapter 2 is almost totally devoted to the difficult proof of Toponogov's theorem. Chapter 3 studies coset spaces with invariant metrics under a transitive Lie group of isometries, in particular Lie groups with bi-invariant metrics and symmetric spaces. These spaces provide a rich source of examples and counter-examples. The curvature is computed efficiently via O'Neill's formula. Chapter 4 summarizes the main results of Morse theory referring to Milnor's excellent book for details. In Chapter 5 one finds results of Klingenberg, Cheeger, Synge, and J. H. C. Whitehead on the cut locus, the injectivity radius, the length of shortest closed geodesics, and the convexity radius. Proofs are often quite geometric, short, and beautiful. The last four chapters (6–9) form the ``core'' of the authors' study. Chapter 6 deals with the sphere theorem (Rauch, Klingenberg, Berger), the maximal diameter theorem (Toponogov), and the minimal diameter theorem (Berger). Chapter 7 studies the differential sphere theorem and closes with a discussion of general problems concerning compact manifolds of positive curvature. The concept of a soul S, a compact totally geodesic and totally convex submanifold, is central to the study of complete (non-compact) manifolds M of nonnegative curvature treated in Chapter 8. In case codim S=1, it is shown that M is isometric to the normal bundle over the soul. A splitting theorem (splitting off a Euclidean factor) is used to obtain a similar result if dimS=1. Another application of the splitting theorem shows for KM≥0 there exists a finite normal subgroup ϕπ1(M) such that π1(M)/ϕ is a Bieberbach group. Chapter 9 studies compact manifolds of non-positive curvature giving some results on the fundamental group. For example, if M is compact, KM≤0, and π1(M)=Γ contains a solvable subgroup Σ, then Σ is a Bieberbach group and M contains the compact flat manifold E/Σ (where E is Euclidean space). Thus π1(M) has a solvable subgroup of finite index if and only if M is flat. There is also Preismann's result: if KM<0, every abelian subgroup of π1(M) is infinite cyclic. Finally, if KM≤0 and π1(M)=Γ1×Γ2 has no center, then M splits isometrically as M1×M2 with π1(Mi)=Γi (Gromoll and Wolf).
Reviewer: Hicks, N. J. [form MathSciNet]