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Riemannian geometry: a modern introduction. (English) Zbl 0810.53001
Cambridge Tracts in Mathematics. 108. Cambridge: Cambridge University Press. xii, 386 p. (1993).
This modern treatment of Riemannian geometry consists of the following chapters: 1. Riemannian manifolds, 2. Riemannian curvature, 3. Riemannian volume, 4. Riemannian coverings, 5. The kinematic density, 6. Isoperimetric inequalities, 7. Comparison and finiteness theorems. Chapters 1 to 4 and 7 present the basic technics and results of Riemannian geometry, such as: connections, geodesics, curvature, Jacobi fields, conjugate and cut loci, Gauss-Bonnet theory of surfaces, H. E. Rauch’s comparison theorem. Furthermore the book deals with more specialized topics: Chapter 3: R. L. Bishop’s volume comparison theorems; Chapter 4: volume growth of Riemannian coverings, discretization of Riemannian manifolds; Chapter 7: Heintze-Karcher volume comparison theorem for the volume of tubular neighborhoods of submanifolds, Alexandrov-Toponogov triangle comparison theorems, Cheeger’s finiteness theorem. Chapter 5 discusses the kinematic density on the set of geodesics (given through the formalism of classical analytical mechanics), it presents Santalo’s formula and applications to manifolds with no conjugate points. Chapter 6 discusses isoperimetric inequalities (via isoperimetric constants). In particular it presents Gromov’s new proofs of the isoperimetric inequalities in Euclidean space and on spheres. And two inequalities of Buser and Croke, followed by their applications in Kanai’s work on discretizations and isoperimetry.
Every chapter of the book features a Note and Exercise section covering literature, examples, applications, and in particular introductions to topics emerging from the ideas in the main body of the text.
Many of the specialized topics here are treated in book form for the first time. This book completes and enriches the relevant literature.

53-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry
53C65 Integral geometry
53C20 Global Riemannian geometry, including pinching