Modern Differential Geometry of Curves and Surfaces with Mathema....pdf

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Alfred Gray Modern Differential Geometry of Curves and Surfaces with Mathematica R Third Edition by Elsa Abbena and Simon Salamon

i Preface to the Second Edition 1 Modern Differential Geometry of Curves and Surfaces is a traditional text, but it uses the symbolic manipulation program Mathematica. This important computer program, available on PCs, Macs, NeXTs, Suns, Silicon Graphics Workstations and many other computers, can be used very eﬀectively for plotting and comput- ing. The book presents standard material about curves and surfaces, together with accurate interesting pictures, Mathematica instructions for making the pic- tures and Mathematica programs for computing functions such as curvature and torsion. Although Curves and Surfaces makes use of Mathematica, the book should also be useful for those with no access to Mathematica. All calculations mentioned in the book can in theory be done by hand, but some of the longer calculations might be just as tedious as they were for diﬀerential geometers in the 19th century. Furthermore, the pictures (most of which were done with Mathematica) elucidate concepts, whether or not Mathematica is used by the reader. The main prerequisite for the book is a course in calculus, both single variable and multi-variable. In addition, some knowledge of linear algebra and a few basic concepts of point set topology are needed. These can easily be obtained from standard sources. No computer knowledge is presumed. In fact, the book provides a good introduction to Mathematica; the book is compatible with both versions 2.2 and 3.0. For those who want to use Curves and Surfaces to learn Mathematica, it is advisable to have access to Wolfram’s book Mathematica for reference. (In version 3.0 of Mathematica, Wolfram’s book is available through the help menus.) Curves and Surfaces is designed for a traditional course in diﬀerential geom- etry. At an American university such a course would probably be taught at the junior-senior level. When I taught a one-year course based on Curves and Surfaces at the University of Maryland, some of my students had computer experience, others had not. All of them had acquired suﬃcient knowledge of Mathematica after one week. I chose not to have computers in my classroom because I needed the classroom time to explain concepts. I assigned all of the problems at the end of each chapter. The students used workstations, PCs 1This is a faithful reproduction apart from the updating of chapter references. It already incorporated the Preface to the First Edition dating from 1993.

ii and Macs to do those problems that required Mathematica. They either gave me a printed version of each assignment, or they sent the assignment to me by electronic mail. Symbolic manipulation programs such as Mathematica are very useful tools for diﬀerential geometry. Computations that are very complicated to do by hand can frequently be performed with ease in Mathematica. However, they are no substitute for the theoretical aspects of diﬀerential geometry. So Curves and Surfaces presents theory and uses Mathematica programs in a complementary way. Some of the aims of the book are the following. • To show how to use Mathematica to plot many interesting curves and sur- faces, more than in the standard texts. Using the techniques described in Curves and Surfaces, students can understand concepts geometrically by plotting curves and surfaces on a monitor and then printing them. The eﬀect of changes in parameters can be strikingly portrayed. • The presentation of pictures of curves and surfaces that are informative, interesting and accurate. The book contains over 400 illustrations. • The inclusion of as many topics of the classical diﬀerential geometry and surfaces as possible. In particular, the book contains many examples to illustrate important theorems. • Alleviation of the drudgery of computing things such as the curvature and torsion of a curve in space. When the curvature and torsion become too complicated to compute, they can be graphed instead. There are more than 175 miniprograms for computing various geometric objects and plotting them. • The introduction of techniques from numerical analysis into diﬀerential geometry. Mathematica programs for numerical computation and draw- ing of geodesics on an arbitrary surface are given. Curves can be found numerically when their torsion and curvature are speciﬁed. • To place the material in perspective through informative historical notes. There are capsule biographies with portraits of over 75 mathematicians and scientists. • To introduce interesting topics that, in spite of their simplicity, deserve to be better known. I mention triply orthogonal systems of surfaces (Chap- ter 19), Bj¨orling’s formula for constructing a minimal surface containing a given plane curve as a geodesic (Chapter 22) and canal surfaces and cyclides of Dupin as Maxwell discussed them (Chapter 20).

iii • To develop a dialect of Mathematica for handling functions that facilitates the construction of new curves and surfaces from old. For example, there is a simple program to generate a surface of revolution from a plane curve. • To provide explicit deﬁnitions of curves and surfaces. Over 300 Mathematica deﬁnitions of curves and surfaces can be used for further study. The approach of Curves and Surfaces is admittedly more computational than is usual for a book on the subject. For example, Brioschi’s formula for the Gaussian curvature in terms of the ﬁrst fundamental form can be too com- plicated for use in hand calculations, but Mathematica handles it easily, either through computations or through graphing the curvature. Another part of Mathematica that can be used eﬀectively in diﬀerential geometry is its special function library. For example, nonstandard spaces of constant curvature can be deﬁned in terms of elliptic functions and then plotted. Frequently, I have been asked if new mathematical results can be obtained by means of computers. Although the answer is generally no, it is certainly the case that computers can be an eﬀective supplement to pure thought, because they allow experimentation and the graphs provide insights into complex relation- ships. I hope that many research mathematicians will ﬁnd Curves and Surfaces useful for that purpose. Two results that I found with the aid of Mathematica are the interpretation of torsion in terms of tube twisting in Chapter 7 and the construction of a conjugate minimal surface without integration in Chapter 22. I have not seen these results in the literature, but they may not be new. The programs in the book, as well as some descriptive Mathematica note- books, will eventually be available on the web. Sample Course Outlines There is ample time to cover the whole book in three semesters at the under- graduate level or two semesters at the graduate level. Here are suggestions for courses of shorter length. • One semester undergraduate differential geometry course: Chapters 1, 2, 7, 9 – 13, parts of 14 – 16, 27. • Two semester undergraduate differential geometry course: Chapters 1 – 3, 9 – 19, 27. • One semester graduate differential geometry course: Chapters 1, 2, 7 – 13, 15 – 19, parts of 22 – 27. • One semester course on curves and their history: Chapters 1 – 8.

iv • One semester course on Mathematica and graphics Chapters 1 – 6, 7 – 11, parts of 14 – 19, 23, and their notebooks. I have tried to include more details than are usually found in mathematics books. This, plus the fact that the Mathematica programs can be used to elucidate theoretical concepts, makes the book easy to use for independent study. Curves and Surfaces is an ongoing project. In the course of writing this book, I have become aware of the vast amount of material that was well-known over a hundred years ago, but now is not as popular as it should be. So I plan to develop a web site, and to write a problem book to accompany the present text. Spanish, German, Japanese and Italian versions of Curves and Surfaces are already available. Graphics Although Mathematica graphics are very good, and can be used to create Quick Time movies, the reader may also want to consider the following additional display methods: • Acrospin is an inexpensive easy-to-use program that works on even the humblest PC. • Geomview is a program for interactive display of curves and surfaces. It works on most unix-type systems, and can be freely downloaded from http://www.geomview.org • Dynamic Visualizer is an add-on program to Mathematica that allows inter- active display. Details are available from http://www.wolfram.com • AVS programs (see the commercial site http://www.avs.com) have been devel- oped by David McNabb at the University of Maryland (http://www.umd.edu) for spectacular stereo three-dimensional images of the surfaces described in this book. A Perspective Mathematical trends come and go. R. Osserman in his article (‘The Geome- try Renaissance in America: 1938–1988’ in A Century of Mathematics in America, volume 2, American Mathematical Society, Providence, 1988) makes the point that in the 1950s when he was a student at Harvard, algebra dominated mathe- matics, the attention given to analysis was small, and the interest in diﬀerential geometry was converging to zero. It was not always that way. In the last half of the 19th century surface theory was a very important area of mathematics, both in research and teaching. Brill,

v then Schilling, made an extensive number of plaster models available to the mathematical public. Darboux’s Lec¸ons sur la Th ´eorie G ´en ´erale des Surfaces and Bianchi’s Lezioni di Geometria Differenziale were studied intensely. I attribute the decline of diﬀerential geometry, especially in the United States, to the rise of tensor analysis. Instead of drawing pictures it became fashionable to raise and lower indices. I strongly feel that pictures need to be much more stressed in diﬀerential geometry than is presently the case. It is unfortunate that the great diﬀerential geometers of the past did not share their extraordinary intuitions with others by means of pictures. I hope that the present book contributes in some way to returning the diﬀerential geometry of curves and surfaces to its proper place in the mathematics curriculum. I wish to thank Elsa Abbena, James Anderson, Thomas Banchoﬀ, Marcel Berger, Michel Berry, Nancy Blachman, William Bruce, Renzo Caddeo, Eu- genio Calabi, Thomas Cecil, Luis A. Cordero, Al Currier, Luis C. de Andr´es, Mirjana Djori´c, Franco Fava, Helaman Fergason, Marisa Fern´andez, Frank Fla- herty, Anatoly Fomenko, V.E. Fomin, David Fowler, George Francis, Ben Fried- man, Thomas Friedrick, Pedro M. Gadea, Sergio Garbiero, Laura Geatti, Peter Giblin, Vladislav Goldberg, William M. Goldman, Hubert Gollek, Mary Gray, Joe Grohens, Garry Helzer, A.O. Ivanov, Gary Jensen, Alfredo Jim´enez, Raj Jakkumpudi, Gary Jensen, David Johannsen, Joe Kaiping, Ben Kedem, Robert Kragler, Steve Krantz, Henning Leidecker, Stuart Levy, Mats Liljedahl, Lee Lorch, Sanchez Santiago Lopez de Medrano, Roman Maeder, Steen Markvorsen, Mikhail A. Malakhaltsev, Armando Machado, David McNabb, Jos´e J. Menc´ıa, Michael Mezzino, Vicente Miquel Molina, Deanne Montgomery, Tamara Mun- zner, Emilio Musso, John Novak, Barrett O’Neill, Richard Palais, Mark Phillips, Lori Pickert, David Pierce, Mark Pinsky, Paola Piu, Valeri Pouchnia, Rob Pratt, Emma Previato, Andreas Iglesias Prieto, Lilia del Riego, Patrick Ryan, Gia- como Saban, George Sadler, Isabel Salavessa, Simon Salamon, Jason P. Schultz, Walter Seaman, B.N. Shapukov, V.V. Shurygin, E.P. Shustova, Sonya ˇSimek, Cameron Smith, Dirk Struik, Rolf Sulanke, John Sullivan, Daniel Tanr`e, C. Terng, A.A. Tuzhilin, Lieven Vanhecke, Gus Vlahacos, Tom Wickam-Jones and Stephen Wolfram for valuable suggestions. Alfred Gray July 1998

vi Preface to the Third Edition Most of the material of this book can be found, in one form or another, in the Second Edition. The exceptions to this can be divided into three categories. Firstly, a number of modiﬁcations and new items had been prepared by Alfred Gray following publication of the Second Edition, and we have been able to incorporate some of these in the Third Edition. The most obvious is Chapter 21. In addition, we have liberally expanded a number of sections by means of additional text or graphics, where we felt that this was warranted. The second is Chapter 23, added by the editors to present the popular theory of quaternions. This brings together many of the techniques in the rest of the book, combining as it does the theory of space curves and surfaces. The third concerns the Mathematica code presented in the notebooks. Whilst this is closely based on that written by the author and displayed in previous editions, many programs have been enhanced and sibling ones added. This is to take account of the progressive presentation that Mathematica notebooks oﬀer, and a desire to publish instructions to generate every ﬁgure in the book. The new edition does diﬀer notably from the previous one in the manner in which the material is organized. All Mathematica code has been separated from the body of the text and organized into notebooks, so as to give readers interactive access to the material. There is one notebook to accompany each chapter, and it contains relevant programs in parallel with the text, section by section. An abridged version is printed at the end of the chapter, for close reference and to present a fair idea of the programs that ride in tandem with the mathematics. The distillation of computer code into notebooks also makes it easier to conceive of rewriting the programs in a diﬀerent language, and a project is underway to do this for Maple. The full notebooks can be downloaded from the publisher’s site http://www.crcpress.com Their organization and layout is discussed in more detail in Notebook 0 below. They contain no output, as this can be generated at will. All the ﬁgures in the book were compiled automatically by merely evaluating the notebooks chapter by chapter, and this served to ‘validate’ the notebooks using Version 5.1 of Mathematica. It is the editors’ intention to build up an on-line database of solutions to the exercises at the end of each chapter. Those marked M are designed to be solved with the help of a suitable Mathematica program.

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