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Curved Spaces
From Classical Geometries to Elementary Differential Geometry

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textbook
  • Date Published: January 2008
  • availability: Available
  • format: Paperback
  • isbn: 9780521713900

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  • This self-contained 2007 textbook presents an exposition of the well-known classical two-dimensional geometries, such as Euclidean, spherical, hyperbolic, and the locally Euclidean torus, and introduces the basic concepts of Euler numbers for topological triangulations, and Riemannian metrics. The careful discussion of these classical examples provides students with an introduction to the more general theory of curved spaces developed later in the book, as represented by embedded surfaces in Euclidean 3-space, and their generalization to abstract surfaces equipped with Riemannian metrics. Themes running throughout include those of geodesic curves, polygonal approximations to triangulations, Gaussian curvature, and the link to topology provided by the Gauss-Bonnet theorem. Numerous diagrams help bring the key points to life and helpful examples and exercises are included to aid understanding. Throughout the emphasis is placed on explicit proofs, making this text ideal for any student with a basic background in analysis and algebra.

    • A concrete approach to the theory, with emphasis on self-contained explicit proofs; uses the classical geometries to motivate the basic ideas of elementary differential geometry
    • Provides a link between basic undergraduate courses on Analysis and Algebra, and more advanced theoretical courses in geometry
    • Rigorous treatment of the classical geometries, via analytical ideas, with exercises at the end of each chapter, reinforcing the material in the text
    • A novel approach to defining curvature on abstract surfaces, and to proving the topological invariance of the Euler number
    • Coverage of a wide range of topics, starting with very elementary material and concluding with rather more advanced mathematical ideas
    • Certain geometrical themes, such as geodesics, curvature, and the Gauss-Bonnet theorem, running throughout the book, provide a unifying philosophy
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    Reviews & endorsements

    " ...the patient reader will acquire substantial techniques and methods that are part of differential geometry and, along with that, much, much more ... certainly a welcome addition to the literature."
    Mathematical Reviews

    "Curved Spaces provides an elegant, sophisticated treatment of two-dimensional geometries suitable for advanced undergraduates... Overall, Wilson has provided a monograph that could ably serve for an excellent undergraduate capstone experience."
    S.J. Colley, Oberlin College for CHOICE

    "Every chapter is followed by an assortment of helpful examples... the book is remarkably self-contained. On the other hand the author does not shun detailed proofs. All these ingredients make for a successful volume.
    Johann Lang, Zentralblatt MATH

    "The book is certainly a welcome addition to the literature. It is clear to the reviewer that the text is a labor of love."
    Richard Escobales, Jr., Mathematical Reviews

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    Product details

    • Date Published: January 2008
    • format: Paperback
    • isbn: 9780521713900
    • length: 198 pages
    • dimensions: 244 x 170 x 11 mm
    • weight: 0.33kg
    • contains: 79 b/w illus. 105 exercises
    • availability: Available
  • Table of Contents

    Preface
    1. Euclidean geometry
    2. Spherical geometry
    3. Triangulations and Euler numbers
    4. Riemannian metrics
    5. Hyperbolic geometry
    6. Smooth embedded surfaces
    7. Geodesics
    8. Abstract surfaces and Gauss-Bonnet.

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    Curved Spaces

    P. M. H. Wilson

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  • Author

    P. M. H. Wilson, University of Cambridge
    Pelham Wilson is Professor of Algebraic Geometry in the Department of Pure Mathematics, University of Cambridge. He has been a Fellow of Trinity College since 1981 and has held visiting positions at universities and research institutes worldwide, including Kyoto University and the Max-Planck-Institute for Mathematics in Bonn. Professor Wilson has over 30 years of extensive experience of undergraduate teaching in mathematics, and his research interests include complex algebraic varieties, Calabi-Yau threefolds, mirror symmetry, and special Lagrangian submanifolds.

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