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**Riemannian manifolds: an introduction to curvature.**
*(English)*
Zbl 0905.53001

Graduate Texts in Mathematics. 176. New York, NY: Springer. xv, 224 p. (1997).

The book’s aim is to develop tools and intuition for studying the central unifying theme in Riemannian geometry, which is the notion of curvature and its relation with topology. The main ideas of the subject, motivated as in the original papers, are introduced here in an intuitive and accessible way.

The history of the subject begins with the Euclidean plane, followed by the theory of curved surfaces in space due to Gauss and later by the modern theory of manifolds with Riemannian metrics. Two major types of geometric results, “classification theorems” and “local-global theorems”,fre illustrated in the beginning of the book, respectively by the side-side-side theorem and angle-sum theorem from Euclidean triangle geometry. A brief review of some background material about tensors on manifolds and vector bundles is followed by the introduction of the Riemannian metric, which gives the framework of the theory developed further. The results, which are valid in the semi-Riemannian case as well, are pointed out. Geodesics, as a generalization of straight lines, lead to the problem of finding an invariant interpretation of the acceleration of a curve, as a way to motivate the definition of connection, which is introduced on a general vector bundle.

The study of linear connections on manifolds, geodesics and “parallel translation” is followed by the special case of Riemannian connection and its geodesics, the exponential map and normal coordinates. The distance minimizing property of geodesics is studied and the Hopf-Rinow theorem is proved.

The curvature tensor is defined as a measure of the failure of second covariant derivatives to commute and has the qualitative interpretation as obstruction to flatness (i.e., local equivalence to Euclidean space). Hopf’s rotation angle formula is generalized by the Gauss-Bonnet formula, leading to the Gauss-Bonnet theorem, which is the first “local-global” theorem. After a study of Jacobi fields and conjugate points, other “local-global” theorems relating curvature and topology are given: Cartan-Hadamard, Bonnet and Cartan-Ambrose-Hicks.

The book is an excellent introduction designed for a one-semester graduate course, containing exercises and problems which encourage students to practice working with the new notions and develop skills for later use. By citing suitable references for detailed study, the reader is stimulated to inquire into further research.

The history of the subject begins with the Euclidean plane, followed by the theory of curved surfaces in space due to Gauss and later by the modern theory of manifolds with Riemannian metrics. Two major types of geometric results, “classification theorems” and “local-global theorems”,fre illustrated in the beginning of the book, respectively by the side-side-side theorem and angle-sum theorem from Euclidean triangle geometry. A brief review of some background material about tensors on manifolds and vector bundles is followed by the introduction of the Riemannian metric, which gives the framework of the theory developed further. The results, which are valid in the semi-Riemannian case as well, are pointed out. Geodesics, as a generalization of straight lines, lead to the problem of finding an invariant interpretation of the acceleration of a curve, as a way to motivate the definition of connection, which is introduced on a general vector bundle.

The study of linear connections on manifolds, geodesics and “parallel translation” is followed by the special case of Riemannian connection and its geodesics, the exponential map and normal coordinates. The distance minimizing property of geodesics is studied and the Hopf-Rinow theorem is proved.

The curvature tensor is defined as a measure of the failure of second covariant derivatives to commute and has the qualitative interpretation as obstruction to flatness (i.e., local equivalence to Euclidean space). Hopf’s rotation angle formula is generalized by the Gauss-Bonnet formula, leading to the Gauss-Bonnet theorem, which is the first “local-global” theorem. After a study of Jacobi fields and conjugate points, other “local-global” theorems relating curvature and topology are given: Cartan-Hadamard, Bonnet and Cartan-Ambrose-Hicks.

The book is an excellent introduction designed for a one-semester graduate course, containing exercises and problems which encourage students to practice working with the new notions and develop skills for later use. By citing suitable references for detailed study, the reader is stimulated to inquire into further research.

Reviewer: C.-L.Bejan (Iaşi)

### MSC:

53-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry |

53Cxx | Global differential geometry |