As to geometry, in particular, the abstract tendency has here led to the magnificent systematic theories of Algebraic Geometry, of Riemannian Geometry, and of Topology; these theories make extensive use of abstract reasoning and symbolic calculation in the sense of algebra. Notwithstanding this, it is still as true today as it ever was that intuitive understanding plays a major role in geometry. And such concrete intuition is of great value not only for the research worker, but also for anyone who wishes to study and appreciate the results of research in geometry.
— David Hilbert [ SE: Hilbert]
These words were written in 1934 by the "father of Formalism," David Hilbert, from the Preface to Geometry and the Imagination by Hilbert and Cohn-Vossen.
The formalisms of differential geometry are considered by many to be among the most complicated and inaccessible of all the formal systems in mathematics. It is probably fair to say that most mathematicians do not feel comfortable with their understanding of differential geometry. In addition, there is little agreement about which formalisms to use and about how to describe them, with the result that the starting definitions, notations and analytic descriptions vary widely from textbook to textbook. What all of these different approaches have in common are underlying geometric intuitions of the basic notions such as straightness (geodesic), smooth, tangent, curvature, and parallel transport.
In this book we will study a foundation for differential geometry based not on analytic formalisms but rather on these underlying geometric intuitions. This foundation should be accessible to anyone with a flexible geometric imagination. It may then be possible that this foundation will serve as a common starting point for the various analytic formalizations. In this book we will explore some of these analytical formalisms. In addition, this geometric foundation relates more directly with our actual experiences of curves and surfaces both in the physical world and in the context of computer graphics.
I invite the reader to explore the basic ideas of differential geometry. I am interested in conveying a different approach to mathematics, stimulating the reader to take a broader and deeper experience of mathematics. An active participation with these ideas, including exploring and writing about them, will give the reader a broader context and experience which is vital for anyone who wishes to understand differential geometry at a deeper level. More and more of the formal analytical aspects of differential geometry have now been mechanized and this mechanization is widely available on personal computers, but the experience of meaning in differential geometry is still a human enterprise that is necessary for creative work.
I believe that mathematics is a natural and deep part of human experience and that experiences of meanings in mathematics are accessible to everyone. Much of mathematics is not accessible through formal approaches except to those with specialized learning. However, through the use of non-formal experience and geometric imagery, many levels of meaning in mathematics can be opened up in a way that more human beings can experience and find intellectually challenging and stimulating.
This text builds on a foundation of intuitive geometric ideas and then ties them into the formalisms of extrinsic and intrinsic differential geometry. The first chapter is an extensive collection of examples of surfaces which are discussed as much as can be done using elementary techniques and geometric intuition. Many of the concepts in the text are introduced first in Chapter 1. Throughout the text there is an emphasis on looking at curves and surfaces in as many different ways as possible but with a particular emphasis on intrinsic, coordinate-free approaches in order to highlight the geometry.
The discussions in the book were written for undergraduate mathematics majors and thus assume of the reader a corresponding level of interest and mathematical sophistication. Previous experience with multivariable calculus and linear algebra are strongly recommended. There is more material in this text than I cover in one semester, so the instructor can choose to leave out certain topics. To assist in this process some problems or parts of problems are labeled with an asterisk (*) indicating that they are not essential for the remainder of the text.
There are some results in this text which (as far as I know) have never before appeared in print. These include the annular hyperbolic plane (which I learned from William Thurston, see Problems 1.8 and 5.7), the use of zooming and fields of view in defining smoothness (see the beginnings of Chapters 2 and 3, especially Problems 2.1, 2.2, and 3.1), and the ribbon test for a geodesic (Problems 3.4 and 7.5).
There is a unique problem-centered approach in the presentation of this material. The main geometric notions, both theory and concepts, are introduced through problems which are designed to give the students an opportunity to experience their own meanings in the material. This is similar to the approach which has been used successfully in the author's Experiencing Geometry on Plane and Sphere (Prentice-Hall, 1996) . Two discussions describing different aspects of this approach can be found in: [David Henderson, "I Learn Mathematics From My Students — Multiculturalism in Action", For the Learning of Mathematics, v. 16, n.2, pp.46-52.] and [Jane-Jane Lo, Kelly Gaddis and David Henderson, "Learning Mathematics Through Personal Experiences: A Geometry Course in Action", For the Learning of Mathematics, v. 16, n.1, pp.34-40.]
Those readers who have access to computer systems running Maple©, Mathematica©, Derive©, or similar software can use these systems to facilitate gaining geometric intuition and imagination of the concepts of differential geometry. In Appendix C I have included several computer exercises for Maple and these and additional scripts are also available for downloading on-line at http://math.cornell.edu/~dwh/books/dg/title.html. I will also include at this site additonal information and updates as they become available.
I started writing problems such as those that appear in this book while teaching differential geometry in the spring of 1992. Again in the spring of 1994 I wrote more problems and used them together with a published textbook in the course. In the spring of 1995 I taught the course using only my problems and altered them and extended them as we went along. Finally the first preliminary version of this text was available in photocopy form in the fall of 1995. It was used by Brian Mortimer in his differential geometry class at Carleton University in Ottawa, Canada, and by James West in his differential geometry course at Cornell University. Both of these mathematicians gave me valuable feedback. In addition, I received many helpful suggestions and comments from the reviewers of the Fall 1995 version — these reviewers were Brian Mortimer, Bruce Piper (Rensselear Polytechnic Institute), Nicola Garofalo (Purdue University), and four anonymous reviewers including ones from Drew Universtiy and University of California at Berkeley. In addition, I received valuable feedback from Jane-Jane Lo, Dexter Luthulli, Cathy Stenson, and Alex Tsow. From these students, instructors and others I have received encouragement and much valuable feedback that has resulted in what I consider to be a better book. Cathy Stenson wrote the Maple computer exercises.
In October 1995, I gave a copy of the preliminary text to Daina Taimiða, faculty member of the Department of Mathematics and Physics at the University of Latvia in Riga, who gave me more feedback. In the summer of 1996, during a two-months visit to Latvia, I with Professor Taimiða went through the collected comments and suggestions and rewrote and extended the text to its present form.
The entire production of the manuscript (typing, formatting, drawings, and final layout and type-setting) has been accomplished using AmiPro, an integrated word processing software. Finally, I wish to thank George Lobell, Senior Editor at Prentice-Hall, for the vision and enthusiasm with which he shepherded this book through the publication process.
Ithaca, NY, February 1997
David W. Henderson
Instead pay attention to meanings behind the words.
But, do not just pay attention to meanings behind the words;
Instead pay attention to your deep experience of those meanings.
This quote expresses the philosophy upon which this book is based. This book will present you with a series of problems. You should explore each question and write out your thinking in a way that can be shared with others. By doing this you will be able to actively develop ideas prior to passively reading or listening to comments of others. When working on the problems, you should be open-minded and flexible and let your thinking wander. Some problems will have short, fairly definitive answers, and others will lead into deep areas of meaning which can be probed almost indefinitely. You should not accept anything just because you remember it from school or because some authority says it is good. Insist on understanding (or seeing) why it is true or what it means for you. Pay attention to your deep experience of these meanings.
You should think about the problems and express your thinking about them even when you know you cannot do them completely. This is important because:
Throughout the text there is an emphasis on looking at curves and surfaces in as many different ways as possible but with a particular emphasis on intrinsic, coordinate-free approaches in order to highlight the geometry. For those readers who wish to avoid local coordinates may, in general, do so by leaving out the problems and sections that refer to them. From the beginning through Problem 4.7 local coordinates are only used as examples. Local coordinates and the associated formalisms are needed in a crucial way only in Problem 4.8 and in the problems following, 6.1.
Those readers who have access to computer systems running Maple©, Mathematica©, Derive©, or similar software can use these systems to facilitate gaining geometric intuition and imagination of the concepts of differential geometry. In Appendix C we have included several computer exercises for Maple and these and additional scripts are also available for downloading on-line at http://math.cornell.edu/~dwh/books/dg/title.html. Similar scripts for other programs should be easily constructed. These exercises are labeled according to the problem in the text to which they are most applicable. They are also referenced at appropriate points in the text. However, the current state-of-the-art for generally available computer graphing programs is not capable of producing what would be the most useful displays. For example, it is not currently possible, with widely useable programs, to view a curve on a surface and to use the mouse to dynamically move a point along the curve and see displayed the three curvature vectors — intrinsic (geodesic), extrinsic, and normal. I hope that interested readers will add to the available collection of scripts by sending to the author (dwh@math.cornell.edu) their scripts or URL's to where their scripts are available on the WWW. Please also use this same e-mail address to send to the author any comments about the book or its use.