Customer Reviews

Differential Geometry: Curves - Surfaces - Manifolds, Second Edition

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This is a very fast moving book, covering a huge amount of material at a fairly sophisticated level in under 380 pages. For example, differential forms are introduced in about 2 pages so that the Maurer-Cartan structural equations can be defined. The first 4 chapters makes up a very concise course in curves and surfaces, while the last 4 chapters cover Riemannian geometry. In comparison, do Carmo's two books take 500 pages for the former and 320 pages for the latter.

For this reason I think the claim that this could be used as an undergraduate text is overly optimistic. For that I would use a more self-contained text like Millman & Parker (ISBN: 0132641437). But it would make an excellent text for a graduate survey, or as a second text for someone wanting to make the transition from classical theory (learned from, say, one of the Dover books like Struik, ISBN: 0486656098) to more modern methods. Also, you'll probably want to supplement with a gentler introduction to differential forms.

Of interest to students of physics, the book covers curves and surfaces in Minkowski space, as well as Einstein spaces.

While there is exist many classic texts on differential geometry, I have particularly appreciated this book for its up-to-date treatment, numerous well-done figures, broad coverage, elegant type-setting, and clear expositions. The book covers all the basics expected from an introduction to differential geometry, including curves and 2-D surfaces, but with a look towards the more advanced material in the second half of the book. It alternates between Ricci style notation and Koszul style notation, often carefully explaining the relation between the two and giving examples (I found this particularly helpful). There are, however, some sections where the english is a bit rough (perhaps the fault of the translator). It is also quite brisk throughout, often mentioning advanced topics before they are treated in detail. For example, it already mentions submanifolds, tangent spaces, and tangent bundles in the first chapter on "Notations and Prerequisites from Analysis." It will require serious attention, especially if one has not encountered a good dose of abstract mathematics before. Nonetheless, I have found myself returning to it over several years as an excellent reference and source of many additional topics that I skipped on a first reading. For example, the final chapter on Einstein spaces is a valuable, though demanding, bonus. Thanks to the AMS for publishing a fine edition of a top-notch German author's work.

This book is very useful for students who are interested in geometry. The book is organized from elementary facts to advanced geometry very well. This book provides to students the reason why they study the geometry. This book explains very easily that the geometry of curves and surfaces can be generalized to high dimensional Riemannian manifolds naturally.
Moreover, the edition of this book is very beautiful and helpful for readers. For example, the important results are placed in boxes.

Differential geomety is perhaps the most beautiful part of higher mathematics. It combines geometry, analysis and intuition in a wonderful way. This attractive book is a concise and modern book that manages to be both pedagogical and accurate in a pleasant way. In only 350 pages most of the differential geometry that a non-expert will ever need is outlined. Illustrations and notation seem optimal for their purpose. The book is a worthy successor of classics like Struik, Stoker, and Kreyzig.

The author gives a clean and wise introduction to the three major parts in differential geometry-curves-surfaces-manifolds. The important concepts in classic results were introduced by short but fully content paragraphs.

The author wrote no gossip in the context and always touch the ideas with a niddle; therefore I should follow that:

This is the best book for introducing differential geometry.