Applications of Gauss-Bonnet.Ĭhapter 8: Global Structures of Surfaces: Length-Minimizing Properties of Geodesics. Congruence of Surfaces.Ĭhapter 7: Riemannian Geometry: Geometric Surfaces. Andrew Pressley is Professor of Mathematics at Kings College London, UK. Differentiable Functions and Tangent Vectors. Elementary Differential Geometry presents the main results in the differential. Nowadays he's an old fogey who teaches at Colby College in Waterville, ME.Įuclidean Space. Gouvêa read parts of O'Neill's book way back when he was an undergraduate. But I do know that O'Neill's book is very nice, the sort of book that can make the right student learn to love the subject.įernando Q. I don't know which I prefer it depends on the day. His competitors (e.g., the books by do Carmo, Oprea, and Pressley) all prefer to avoid differential forms. Whew The change that pleases me the most is that the new edition makes a little more use of linear algebra.
O'Neill has remained alone in choosing this approach. 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. On the other hand, this is the language used in more advanced books, so learning it can be very useful to the student. This can get in the way of the geometry, as I learned when I tried to teach from this book. There is no doubt that this is the right language, but using it in an introductory book means quite a large amount of setup is needed right at the beginning of the course. As the author points out, it is instructive to do these calculations by hand once or twice, but once the matter is understood one should use technology.įor instructors looking for an undergraduate textbook, the most problematic aspect of this book will be O'Neill's decision to use the language of tangent vectors, differential forms, and covariant derivatives.
ELEMENTARY DIFFERENTIAL GEOMETRY PRESSLEY HOW TO
Beyond that, the only real change is the addition of a very useful appendix showing how to use Mathematica or Maple to do the more important computations in elementary differential geometry. There are lots of corrections, and the exercises have been significantly improved. The changes to the text are fairly minimal. It's good to see it come back in a new edition. But it's still a beautiful book, well-written and displaying a deep understanding of the subject. And with old-fashioned artist-drawn illustrations. Forty years later, O'Neill has lots of competition, of course. There was simply no other book taking a modern approach to the material that could be used as an undergraduate text. At that point, it pretty much had the field to itself.
Barrett O'Neill's introduction to differential geometry was first published in 1966.
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