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Metric Structures in Differential Geometry : Graduate Texts in Mathematics - Gerard Walschap

Metric Structures in Differential Geometry

Graduate Texts in Mathematics

Hardcover

Published: 18th March 2004
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This text is an introduction to the theory of differentiable manifolds and fiber bundles. The only requisites are a solid background in calculus and linear algebra, together with some basic point-set topology. The first chapter provides a comprehensive overview of differentiable manifolds. The following two chapters are devoted to fiber bundles and homotopy theory of fibrations. Vector bundles have been emphasized, although principal bundles are also discussed in detail. The last three chapters study bundles from the point of view of metric differential geometry: Euclidean bundles, Riemannian connections, curvature, and Chern-Weil theory are discussed, including the Pontrjagin, Euler, and Chern characteristic classes of a vector bundle. These concepts are illustrated in detail for bundles over spheres. Chapter 5, with its focus on the tangent bundle, also serves as a basic introduction to Riemannian geometry in the large. This book can be used for a one-semester course on manifolds or bundles, or a two-semester course in differential geometry.

Gerard Walschap is Professor of Mathematics at the University of Oklahoma where he developed this book for a series of graduate courses he has taught over the past few years.

From the reviews:

"The book gives an introduction to the basic theory of differentiable manifolds and fiber bundles ... The book is well written. The presentation is clear, detailed and essentially self-contained. This book is suitable for senior undergraduate and graduate students. It can be used for a course on manifolds and bundles, or a course in differential geometry." (M. Burkhardt, Zeitschrift fur Analysis und ihre Anwendungen, 1, 2005)

"This text is an introduction to the theory of differentiable manifolds and fiber bundles ... provides a comprehensive overview of differentiable manifolds ... concepts are illustrated in detail for bundles over spheres ... This book can be used for a one-semester course on manifolds or bundles, or a two-semester course in differential geometry." (L'enseignement mathematique, 50:1-2, 2004)

"This book is based on the author's graduate-level lecture notes. ... One of the strengths of this book is the fact that the author manages in a 220-page volume to cover important themes in Riemannian geometry and fiber bundles. ... The book contains some nice examples ... . The topics are well-closed and the content is well-organized. ... This clearly written book is an excellent source for teaching a course in differential geometry ... . It is a worthwhile addition to any mathematical library." (Stere Ianus, Zentralblatt MATH, Vol. 1083, 2006)

"This text should be an elementary introduction to differential geometry. ... The style is rather concise and many facts are shifted to 165 nontrivial exercises. The book is very well written and can be recommended to those who want to learn the topic quickly and actively." (EMS Newsletter, June, 2005)

"This book is a carefully written text for an introductory graduate course on differentiable manifolds, fiber bundles and Riemannian geometry. ... This book is a thorough and insightful introduction to modern differential geometry with many interesting examples and exercises that illustrate key concepts effectively; it is highly recommended by the reviewer." (Thomas E. Cecil, Mathematical Reviews, Issue 2006 e)

"In every mathematical library a number of introductory books to differential geometry can be found. They are all different in some aspect, but - at the same time - none presents all the concepts equally successfully to all the readers. So there is allways a need for new introductory books, and Walschap's book is a good one of these. ... The series of definitions, concepts and theories are punctuated by examples, remarks. Each section ends with exercises." (Arpad Kurusa, Acta Scientiarum Mathematicarum, Vol. 73, 2007)

Preface
Differentiable Manifolds
Basic Definitions
Differentiable Maps
Tangent Vectors
The Derivative
The Inverse and Implicit Function Theorems
Submanifolds
Vector Fields
The Lie Bracket
Distributions and Frobenius Theorem
Multilinear Algebra and Tensors
Tensor Fields and Differential Forms
Integration on Chains
The Local Version of Stokes' Theorem
Orientation and the Global Version of Stokes' Theorem
Some Applications of Stokes' Theorem
Fiber Bundles
Basic Definitions and Examples
Principal and Associated Bundles
The Tangent Bundle of Sn
Cross-Sections of Bundles
Pullback and Normal Bundles
Fibrations and the Homotopy Lifting/Covering Properties
Grassmannians and Universal Bundles
Homotopy Groups and Bundles Over Spheres
Differentiable Approximations
Homotopy Groups
The Homotopy Sequence of a Fibration
Bundles Over Spheres
The Vector Bundles Over Low-Dimensional Spheres
Connections and Curvature
Connections on Vector Bundles
Covariant Derivatives
The Curvature Tensor of a Connection
Connections on Manifolds
Connections on Principal Bundles
Metric Structures
Euclidean Bundles and Riemannian Manifolds
Riemannian Connections
Curvature Quantifiers
Isometric Immersions
Riemannian Submersions
The Gauss Lemma
Length-Minimizing Properties of Geodesics
First and Second Variation of Arc-Length
Curvature and Topology
Actions of Compact Lie Groups
Characteristic Classes
The Weil Homomorphism
Pontrjagin Classes
The Euler Class
The Whitney Sum Formula for Pontrjagin and Euler Classes
Some Examples
The Unit Sphere Bundle and the Euler Class
The Generalized Gauss-Bonnet Theorem
Complex and Symplectic Vector Spaces
Chern Classes
Bibliography
Index
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9780387204307
ISBN-10: 038720430X
Series: Graduate Texts in Mathematics
Audience: Tertiary; University or College
Format: Hardcover
Language: English
Number Of Pages: 229
Published: 18th March 2004
Publisher: Springer-Verlag New York Inc.
Country of Publication: US
Dimensions (cm): 22.9 x 15.2  x 1.91
Weight (kg): 1.15