Differential Geometry of Curves and Surfaces

1. Front Matter

   Pages i-xii

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2. Plane Curves and Space Curves

   • Shoshichi Kobayashi

   Pages 1-34

3. Local Theory of Surfaces in the Space

   • Shoshichi Kobayashi

   Pages 35-75

4. Geometry of Surfaces

   • Shoshichi Kobayashi

   Pages 77-108

5. The Gauss-Bonnet Theorem

   • Shoshichi Kobayashi

   Pages 109-131

6. Minimal Surfaces

   • Shoshichi Kobayashi

   Pages 133-156

7. Correction to: Differential Geometry of Curves and Surfaces

   • Shoshichi Kobayashi

   Pages C1-C1

8. Back Matter

   Pages 157-192

   PDF

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. 

• curves
• surfaces
• curvature
• Riemannian metric
• Gauss--Bonnet's theorem
• minimal surfaces

“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)

• University of California, Berkeley, USA

  Shoshichi Kobayashi

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. 

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