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Elementary Differential Geometry

Andrew Pressley
Publication Date: 
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Springer Undergraduate Mathematics Series
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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Fernando Q. Gouvêa
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When a new edition of a book I know and love is announced, I get a little nervous. It’s all too common for authors to “improve” a book in ways that make it much less interesting and much more like all the other textbooks, thereby losing most of what made the first edition distinctive and interesting. There is always a temptation, as well, to add stuff — sometimes lots of stuff — requested by users of the book, all of whom, it seems, would like their pet topic to be included.

In the light of that, I am very happy to report that the new edition of Pressley’s Elementary Differential Geometry is an even better book than the first edition, which I reviewed some time ago. Whew!

The change that pleases me the most is that the new edition makes a little more use of linear algebra. In the first edition, the Weingarten operator was never defined, with the result that a certain matrix played a very important — but entirely unexplained — role. More generally, in the first edition there was no use of the idea of the matrix of a linear transformation with respect to a chosen basis, making it harder to say certain things. In this version, the tangent plane is treated as a vector space, and we can therefore talk about operators and quadratic forms on it. I think this is great. At least in the U.S., every student taking this course is likely to have taken a linear algebra course and encountered these ideas, and every chance to reinforce and use them should be seized. In addition, this way of thinking is just easier!

The other positive change is the inclusion of a discussion of parallel transport. The book still approaches curvature in an elementary way, without introducing connections. But parallel transport is an important idea, and it’s the way most physicists think about curvature, so students should be exposed to it.

There is also some reorganization. Recognizing that instructors can have different things they want to get to in the course, the author has organized the book so that the first ten chapters contain the core material, while the final three chapters present a choice of “culminating result” one could aim for: hyperbolic geometry, minimal surfaces, and the Gauss-Bonnet theorem.

The largest addition to the book is the chapter on hyperbolic geometry. This is something I could do without, but as long as it’s segregated in a chapter of its own, I can safely ignore it. A section on spherical geometry has also been added, which the author suggests should only be covered if the instructor is aiming for hyperbolic geometry as the final topic. To my relief, the elegant treatment of Archimedes’ theorem, which does not depend on doing any detailed spherical geometry, has been preserved.

One thing that some instructors have had trouble with has not changed: there are still full solutions to all problems given in an appendix. To accommodate those of us who need to assign problem sets for credit, however, Pressley has divided his problem sets into two portions. Most of the problems are in the book and have solutions in the back. But there are more problems available on the book’s web site, and solutions are available only to instructors adopting the book. (Many of these, one must note, are problems that were in the first edition with full solutions.)

For what it’s worth, when I used the book I sidestepped the issue by requiring that students write detailed and well-explained solutions in clear, grammatical prose. The solutions given in the book are terse enough that for students to rewrite them to my specs requires actually understanding them… and that’s almost as good as solving the problem on your own. In fact, sometimes it’s harder.

The upshot is that this is still an excellent book and still my first choice for an undergraduate introduction to differential geometry.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME. He teaches differential geometry about once a decade.

Curves in the plane and in space.- How much does a curve curve?.- Global properties of curves.- Surfaces in three dimensions.- Examples of surfaces.- The first fundamental form.- Curvature of surfaces.- Gaussian, mean and principal curvatures.- Geodesics.- Gauss’ theorema egregium.- Hyperbolic geometry.- Minimal surfaces.- The Gauss–Bonnet theorem.- Inner product spaces and self-adjoint linear maps.- A1. Isometries of euclidean spaces.- A2. Möbius transformations.- Hints to selected exercises.- Solutions