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Riemannian Manifolds: An Introduction to Curvature
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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
Links to Related Works
Subject References:
Differential Geometry
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Mathematics
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Authors:
Lee, John M. 1950-
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Lee, John M., 1950-
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Corporate Authors:
SpringerLink (Online service)
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Series:
Graduate texts in mathematics
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GTM
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Classification:
516.36
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