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An Introduction to Differential Geometry

Dover Publications
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I had occasion very recently to teach differential geometry to a rather strong group of seniors (with a precocious sophomore and junior in the mix). I set myself the task of getting them to the point where they had some idea about the behavior of smooth manifolds: the tangent and cotangent spaces, differential forms, and so on. I found that the introductory text I used, while excellent in many other ways, spent too much time on the analysis of curves in the plane and in 3-space, and then surfaces in 3-space, for my taste and purposes. Now I acknowledge that that sort of thing has great merit, given the fact that a wealth of geometrical insight based on low-dimensional examples is a great virtue — it is indispensable for a geometer, I am sure. But I wanted a different route, really, one focused more on conveying the subject’s structure as soon as possible (and within the orbit of one semester); a colleague who sat in on the course characterized my approach as a “from the top down” affair. All in all, then, I think I should have chosen another book.

Another (but connected) aspect one should consider in structuring a differential geometry course at the indicated level is the question of whether to go the route of manifolds per se (as I did) or to place a greater emphasis on (proto-) Riemannian geometry. I speculate that most geometers would opt for the latter. When I was looking for a text for the aforementioned course I came across a number of books with such an orientation, but as I already indicated, they were not in keeping with my desire to get at manifolds. However, granting a desire to get at Riemannian geometry swiftly and effectively, the book under review races to the front of the line.

It is an older book, and a British production, meaning that there are no glitzy pictures on every other page, no computer applets (is that the right word?), and the prose is old-fashioned academese (certainly not the right word, but you get my drift). All in all it is typical of the textbooks coming out of Oxford and Cambridge in the middle part of the last century. I, for one, love these books, but they’re not every one’s cup of tea, as the British might put it.

But there is a lot to love in Willmore’s An Introduction to Differential Geometry, including its profound thoroughness and attention paid to detail. The reader of this book will emerge with a true sense of real differential geometry, including a treatment of tensors consonant with how things are done in, for instance, general relativity, with a strong push in the direction of Riemannian geometry.

This also means that the more overt commutative algebraic aspects are absent, or at least hidden in the shadows. There is no explicit development of the tangent and cotangent space as dual vector spaces, i.e. no general modern manifold theory, which is perhaps explainable by the fact that the revolution in algebraic geometry had not yet happened at the time Willmore wrote the book. On the other hand, the idea might have been to hit Riemannian geometry as hard as possible, early, and that’s that.

Indeed Willmore says as much in his Introduction: “Part 2 introduces the idea of a tensor, first in algebra and the in calculus. It gives the basic ideas of the absolute calculus and the fundamentals of Riemannian geometry.” A little earlier he states: “The book also gives a useful introduction to the methods of differential geometry or to tensor calculus for research students (e.g. in Physics of Engineering) who may wish to apply them.” Clearly that really gives the thrust of what the second part of the book is about. But don’t look for such things as the phrase, “dividing out by bilinearity,” when tensors are introduced: linear algebra qua algebra is pretty much done sub rosa, much along the lines of what the physicists do.

Part 1 of the book “is devoted to the classical theory of curves and surfaces, vector methods being used throughout,” as already indicated above, and that is certainly how things should be. But there’s more: Willmore notes that “[t]he last chapter of Part 1 dealing with the global differential geometry of surfaces contains material which does not appear in any standard English text. Here the student is introduced to differential geometry in the large. Although attention is confined to two-dimensional surfaces, many of the concepts involved can be easily extended to apply to n-dimensional manifolds …”

To be sure, given that the foregoing was written in 1959, the paucity of the indicated English-language material Willmore mentions is no longer there, but it is instructive to consider what he did in the way of breaking new ground, and it is obviously pedagogically sound to work in two dimensions (submersed in three, if needed) before n-manifolds are introduced in their full glory.

In summing things up, then, true to Oxbridge form, the book is scholarly, terse, and deep, even if a bit dated. There are exercises at the end of chapters, of which there are eight — half in Part 1, half in Part 2. We start with space curves; we end with tensor methods in surface theory. Factoring in that tensor calculus on Riemannian manifolds is nasty, what with indices everywhere and all sorts of conventions to remember (consider e.g. the tricky business of covariant differentiation), the book is wonderful and yields great dividends for the reader, the lecturer, and the student, especially for those of us who like the style of our cousins from “across the Pond.”

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.

Date Received: 
Thursday, March 29, 2012
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T. J. Willmore
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Michael Berg

I. The Theory of Space curves
II. The Metric: Local Intrinsic Properties of a Surface
III. The Second Fundamental Form: Local Non-Intrinsic Properties of a Surface
IV. Differential Geometry of Surfaces in the Large
V. Tensor Algebra
VI. Tensor Calculus
VII. Riemannian Geometry
VIII. Applications of Tensor Methods to Surface Theory
Suggestions for Further Reading
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Friday, July 27, 2012