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.
