Overview
- 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
Part of the book series: Springer Undergraduate Mathematics Series (SUMS)
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Table of contents(6 chapters)
About this book
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.
Reviews
“There is a wealth of excellent text books on the differential geometry of curves and surfaces. A rare jewel among them is the recent translation of a Japanese classic written by Shoshichi Kobayaschi … . This volume is a superb addition to the current literature on the geometry of curves and surfaces, and it is of major interest for classroom study, as well for general use as a reference and eventually for self-study.” (Bogdan D. Suceavă, The Mathematical Intelligencer, Vol. 44 (1), March 2022)
“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)
Authors and Affiliations
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University of California, Berkeley, USA
Shoshichi Kobayashi
About the author
Bibliographic Information
Book Title: Differential Geometry of Curves and Surfaces
Authors: Shoshichi Kobayashi
Translated by: Eriko Shinozaki Nagumo, Makiko Sumi Tanaka
Series Title: Springer Undergraduate Mathematics Series
DOI: https://doi.org/10.1007/978-981-15-1739-6
Publisher: Springer Singapore
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer Nature Singapore Pte Ltd. 2019
Softcover ISBN: 978-981-15-1738-9Published: 25 November 2019
eBook ISBN: 978-981-15-1739-6Published: 13 November 2019
Series ISSN: 1615-2085
Series E-ISSN: 2197-4144
Edition Number: 1
Number of Pages: XII, 192
Number of Illustrations: 1 b/w illustrations
Topics: Differential Geometry, Analysis, Manifolds and Cell Complexes (incl. Diff.Topology)