The first two cover curves and surfaces, respectively, in three-space (and sometimes in the plane). Most books with titles like this offer similar content. Yet, there must still be some market for books like this, because several have recently appeared, including a second edition of Differential Geometry of Curves and Surfaces by Banchoff and Lovett and another book with the same title by Kristopher Tapp. Of late, however, it seems to me (based on anecdotal evidence garnered from a highly unscientific survey) that not as many departments offer such a course. I now find myself in the position of once again thanking them, this time for publishing not just a re-issuance, but an actual new edition, of do Carmo’s classic textbook, first published by Prentice-Hall (now part of Pearson) in 1976.īack in the day, it was fairly common for undergraduate mathematics departments to offer a course in differential geometry, which I suppose I should now refer to as “classical” differential geometry (curves and surfaces in the plane and three-space) to distinguish it from “modern” differential geometry (the study of differentiable manifolds). “The book reaches admirable destinations with few formal prerequisites and contains enough material for a leisurely one-semester undergraduate course.For some years now, I, as well as a number of other contributors to this column, have on occasion expressed appreciation to Dover Publications for the service it provides to the mathematical community by re-issuing classic textbooks and making them available to a new generation at an affordable price. 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 entire material is carefully developed, a lot of beautiful examples supporting the understanding. “This is an excellent book written in a clear and precise style. Suceavă, The Mathematical Intelligencer, Vol. 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. A rare jewel among them is the recent translation of a Japanese classic written by Shoshichi Kobayaschi …. “There is a wealth of excellent text books on the differential geometry of curves and surfaces. 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. Chapter 5, Minimal Surfaces, requires some elementary knowledge of complex analysis.
Here again, many illustrations are provided to facilitate the reader’s understanding. 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).
The theorem is a most beautiful and deep result in differential geometry. Then the Gauss–Bonnet theorem, the major topic of this book, is discussed at great length. Chapter 4 starts with a simple and elegant proof of Stokes’ theorem for a domain. The concept of a geodesic introduced in Chapter 2 is extensively discussed, and several examples of geodesics are presented with illustrations. In Chapter 3, the Riemannian metric on a surface is introduced and properties determined only by the first fundamental form are discussed. 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. Two types of curvatures - the Gaussian curvature K and the mean curvature H -are introduced.
Chapter 2 deals with local properties of surfaces in 3-dimensional Euclidean space. Minimal Surfaces.Ĭhapter 1 discusses local and global properties of planar curves and curves in space. Berkeley for 50 years, recently translated by Eriko Shinozaki Nagumo and Makiko Sumi Tanaka. This book is a posthumous publication of a classic by Prof.
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