Riemannian Manifolds : An Introduction to Curvature / by John M. Lee.

Format:

Book

Author/Creator:

Lee, John M., 1950- author.

Contributor:

SpringerLink (Online service)

Series:

Graduate texts in mathematics 0072-5285 ; 176.

Graduate Texts in Mathematics, 0072-5285 ; 176

Language:

English

Subjects (All):

Geometry, Differential.

Differential Geometry.

Local Subjects:

Differential Geometry.

Physical Description:

1 online resource (XV, 226 pages).

Edition:

First edition 1997.

Contained In:

Springer Nature eBook

Place of Publication:

New York, NY : Springer New York : Imprint: Springer, 1997.

System Details:

text file PDF

Summary:

This book is designed as a textbook for a one-quarter or one-semester graduate course on Riemannian geometry, for students who are familiar with topological and differentiable manifolds. It focuses on developing an intimate acquaintance with the geometric meaning of curvature. In so doing, it introduces and demonstrates the uses of all the main technical tools needed for a careful study of Riemannian manifolds. The author has selected a set of topics that can reasonably be covered in ten to fifteen weeks, instead of making any attempt to provide an encyclopedic treatment of the subject. The book begins with a careful treatment of the machinery of metrics, connections, and geodesics,without which one cannot claim to be doing Riemannian geometry. It then introduces the Riemann curvature tensor, and quickly moves on to submanifold theory in order to give the curvature tensor a concrete quantitative interpretation. From then on, all efforts are bent toward proving the four most fundamental theorems relating curvature and topology: the Gauss-Bonnet theorem (expressing the total curvature of a surface in term so fits topological type), the Cartan-Hadamard theorem (restricting the topology of manifolds of nonpositive curvature), Bonnet's theorem (giving analogous restrictions on manifolds of strictly positive curvature), and a special case of the Cartan-Ambrose-Hicks theorem (characterizing manifolds of constant curvature). Many other results and techniques might reasonably claim a place in an introductory Riemannian geometry course, but could not be included due to time constraints.

Contents:

What Is Curvature?

Review of Tensors, Manifolds, and Vector Bundles

Definitions and Examples of Riemannian Metrics

Connections

Riemannian Geodesics

Geodesics and Distance

Curvature

Riemannian Submanifolds

The Gauss-Bonnet Theorem

Jacobi Fields

Curvature and Topology.

Other Format:

Printed edition:

ISBN:

9780387227269

Access Restriction:

Restricted for use by site license.