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Riemannian Manifolds: An Introduction to Curvature

Riemannian Manifolds: An Introduction to Curvature
Catalogue Information
Field name Details
Dewey Class 516.36
Title Riemannian Manifolds ([EBook]) : An Introduction to Curvature / by John M. Lee.
Author Lee, John M. , 1950-
Other name(s) SpringerLink (Online service)
Publication New York, NY : Springer , 1997.
Physical Details XV, 226 pages : online resource.
Series Graduate texts in mathematics 0072-5285 ; ; 176
ISBN 9780387227269
Summary Note 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 note 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.
System details note Online access to this digital book is restricted to subscription institutions through IP address (only for SISSA internal users)
Internet Site http://dx.doi.org/10.1007/b98852
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