Differential Geometry of Curves and Surfaces by Shoshichi Kobayashi, translated from the Japanese by Eriko Shinozaki Nag
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here is a wealth of excellent textbooks on the differential geometry of curves and surfaces. A rare jewel among them is the recent translation of a Japanese classic written by Shoshichi Kobayashi (1932– 2012), an eminent authority in the field [5, 6, 8]. Requiring as background, according to the author, only elementary calculus and ‘‘matrices of size 2 and 3,’’ the book introduces very early in the presentation the method of moving frames, then gently guides the reader all the way to the Gauss–Bonnet theorem, a goal that can be reached in one semester. Kobayashi encourages the novice to proceed directly to abstract manifolds of arbitrary dimension, as opposed to many other textbooks where hypersurfaces in Rnþ1 are discussed as an intermediary step. The book has five chapters: 1. Plane Curves and Space Curves, 2. Local Theory of Surfaces in the Space, 3. Geometry of Surfaces, 4. The Gauss–Bonnet Theorem, and 5. Minimal Surfaces. Chapter 1 begins with a description of planar curves, which allows the author to introduce the idea of curvature using a planar version of the Frenet–Serret equations. Indeed, in equation (1.2.22), the author obtains that for an orthonormal frame e1 ðsÞ; e2 ðsÞ at a point p(s) of a curve with arc length s, we have
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e10 ðsÞ ¼ jðsÞe2 ðsÞ: In such a way, jðsÞ appears naturally from the differentiation process in the 2-dimensional plane. This way of introducing the idea of curvature is different from, for example, [2, p. 16], where curvature is introduced as the magnitude of acceleration when the curve is parameterized by arc length. This fundamental difference in approaches reveals Kobayashi’s desire to emphasize from the beginning the importance of the method of moving frames, in the tradition of E´lie Cartan (Kentaro Yano was a doctoral student of E´lie Cartan’s before becoming Kobayashi’s teacher and mentor), an idea that has considerable depth and many applications in modern differential geometry. It really matters how this fundamental definition is presented to a motivated audience encountering the topic for the first time.
Particularly well done and well illustrated is the introduction of Gauss’s spherical map, which allows a discussion of the area elements of a surface. Section 2.3 is dedicated to ‘‘Examples and Calculations of Fundamental Forms and Curvatures,’’ while Section 2.6., entitled ‘‘Use of Exterior Differential Forms,’’ contains the first and second structure equations, followed by the Mainardi–Codazzi equations for differential forms. These are described as the fundamental equations of the theory of surfaces. We should point out that while Kobayashi indicates the origin of the method of moving frames to be in the works of Gaston Darboux, on account of his important contribution [3], an important historical remark is made in [1, p. 45], where we learn that the method of moving frames was first used by Martin Bartels (1769–1836), a professor at the University of Dorpat (now Tartu in Estonia). He is best known as a teacher of young Gauss and, later, while at the University of
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