Graduate Texts in Mathematics: Riemannian Manifolds: An Introduction to Curvature (Softcover)

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VERY GOOD: This book is in very good condition, showing very little sign of use or wear. It was printed on acid-free paper.

Product Details

This text is Volume 176 in Springer'sGraduate Texts in Mathematics series.

Continued from the back cover: "The book begins with a careful treatment of the machinery of metrics, connections, and geodesics, and then introduces the curvature tensor as a way of measuring whether a Riemannian manifold is locally equivalent to Euclidean space. Submanifold theory is developed next in order to give the curvature tensor a concrete quantitative interpretation. The remainder of the text is devoted to proving the four most fundamental theorems relating curvature and topology: the Gauss-Bonnet Theorem, the Cartan-Hadamard Theorem, Bonnet's Theorem, and the characterization of manifolds of constant curvature.

"This unique volume will appeal especially to students by presenting a selective introduction to the main ideas of the subject in an easily accessible way. The material is ideal for a single course, but broad enough to provide students with a firm foundation from which to pursue research or develop applications in Riemannian geometry and other fields that use its tools.

"Of special interest are the exercises and problems dispersed throughout the text. The exercises are carefully chosen and timed so as to give the reader opportunities to review material that has just been introduced, to practice working with the definitions, and to develop skills that are used later in the book. The problems that conclude the chapters are generally more difficult. They not only introduce new material not covered in the body of the text, but they also provide the student with indispensable practice in using the techniques explained in the text, both for doing computations and for proving theorems."

Chapters

1. What Is Curvature?
2. Review of Tensors, Manifolds, and Vector Bundles
3. Definitions and Examples of Riemannian Metrics
4. Connections
5. Riemannian Geodesics
6. Geodesics and Distance
7. Curvature
8. Riemannian Submanifolds
9. The Gauss-Bonnet Theorem
10. Jacobi Fields
11. Curvature and Topology

The book concludes with References and Index.