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Springer Undergraduate Mathematics Series

Differential Geometry of Curves and Surfaces

Authors: Kobayashi, Shoshichi

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  • Is the long-awaited English translation of Kobayashi’s classic on differential geometry, acclaimed in Japan as an excellent undergraduate text
    Focuses on curves and surfaces in 3-dimensional Euclidean space, requiring only freshman-level mathematics to understand the celebrated Gauss–Bonnet theorem
    Provides many examples, illustrations, exercise problems with full solutions, and a postscript on the intriguing history of differential geometry

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eBook $34.99
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  • ISBN 978-981-15-1739-6
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Softcover $44.99
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  • ISBN 978-981-15-1738-9
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  • Immediate ebook access, if available*, with your print order
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About this Textbook

This book is a posthumous publication of a classic by Prof. Shoshichi Kobayashi, who taught at U.C. Berkeley for 50 years, recently translated by Eriko Shinozaki Nagumo and Makiko Sumi Tanaka.

There are five chapters: 1. Plane Curves and Space Curves; 2. Local Theory of Surfaces in Space; 3. Geometry of Surfaces; 4. Gauss–Bonnet Theorem; and 5. Minimal Surfaces.

Chapter 1 discusses local and global properties of planar curves and curves in space. Chapter 2 deals with local properties of surfaces in 3-dimensional Euclidean space. Two types of curvatures — the Gaussian curvature K and the mean curvature H —are introduced.  The method of the moving frames, a standard technique in differential geometry, is introduced in the context of a surface in 3-dimensional Euclidean space.  In Chapter 3, the Riemannian metric on a surface is introduced and properties determined only by the first fundamental form are discussed. The concept of a geodesic introduced in Chapter 2 is extensively discussed, and several examples of geodesics are presented with illustrations. Chapter 4 starts with a simple and elegant proof of Stokes’ theorem for a domain.  Then the Gauss–Bonnet theorem, the major topic of this book, is discussed at great length. The theorem is a most beautiful and deep result in differential geometry. It yields a relation between the integral of the Gaussian curvature over a given oriented closed surface S and the topology of S in terms of its Euler number χ(S). Here again, many illustrations are provided to facilitate the reader’s understanding. Chapter 5, Minimal Surfaces, requires some elementary knowledge of complex analysis.  However, the author retained the introductory nature of this book and focused on detailed explanations of the examples of minimal surfaces given in Chapter 2. 

About the authors

Professor Shoshichi Kobayashi was a Professor Emeritus at University of California, Berkeley. He passed away on August 29 in 2012. He was a student of Professor Kentaro Yano at the University of Tokyo. He was one of famous differential geometers not only in Japan but also in the world. He wrote 15 books both in Japanese and in English. 

Reviews

“This is an excellent book written in a clear and precise style. The entire material is carefully developed, a lot of beautiful examples supporting the understanding. This is certainly a book that strongly motivates the reader to continue studying differential geometry, passing from the case of curves and surfaces in 3-dimensional Euclidean space to manifolds.” (Gabriel Eduard Vilcu, zbMATH 1437.53001, 2020)

“The book reaches admirable destinations with few formal prerequisites and contains enough material for a leisurely one-semester undergraduate course.” (MAA Reviews, March 8, 2020)


Table of contents (5 chapters)

Table of contents (5 chapters)

Buy this book

eBook $34.99
price for Mexico
  • ISBN 978-981-15-1739-6
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $44.99
price for Mexico
  • ISBN 978-981-15-1738-9
  • Free shipping for individuals worldwide
  • Immediate ebook access, if available*, with your print order
  • Usually ready to be dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Differential Geometry of Curves and Surfaces
Authors
Translated by
Tanaka, M., Shinozaki, E.
Series Title
Springer Undergraduate Mathematics Series
Copyright
2019
Publisher
Springer Singapore
Copyright Holder
Springer Nature Singapore Pte Ltd.
eBook ISBN
978-981-15-1739-6
DOI
10.1007/978-981-15-1739-6
Softcover ISBN
978-981-15-1738-9
Series ISSN
1615-2085
Edition Number
1
Number of Pages
XII, 192
Number of Illustrations
1 b/w illustrations
Topics

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