# Book Review: O’Neill’s Elementary Differential Geometry, 2nd Ed

Barrett O’Neill’s Elementary Differential Geometry, 2nd Ed

The next book in the stack is Barrett O’Neill’s Elementary Differential Geometry, Revised 2nd Ed. I read a portion of this book in the car today when it was not my turn to drive. Eastern Iowa and western Illinois are visually uninteresting this time of year, so the only distractions were from the kids in the back of the minivan.

This book is published by Academic Press, which is an imprint of Elsevier. Given the ongoing boycott of Elsevier products, I am reluctant to give this a recommendation. If you care about the state of journal publishing (and you should), then you have to think about how much you care about this and whether you want to extend your boycott to products besides journals.

None of that really has much to do with the quality of the book under review, so I’ll soldier on. Oh, you can also get a copy from Amazon.com.

This book is the closest thing I have found to an appropriate text so far. The only concern is the choice to use Cartan’s method of moving frames throughout.

The book is organized under these headings.

• Calculus on Euclidean Space
• Frame Fields
• Euclidean Geometry
• Calculus on a Surface
• Shape Operators
• Geometry of Surfaces in $\mathbb{R}^3$
• Riemannian Geometry
• Global Structure of Surfaces

There is more material here than is typically covered in a one semester course. There is a selection of global theorems in the last chapter, including Gauss-Bonnett, which is the most common conclusion to an undergraduate class.

Keeping with the plan, I read chapter four, Calculus on a Surface. Because I was interested to see how well the choice of exterior calculus would be handled, I also read some parts of chapter one.

### Observations

The text is structure to introduce the method of moving frames (repère mobile) due to Élie Cartan. This means that students have to relearn vector calculus in a way that reflects the language of frame fields and differential forms. From my experience with the course last time, the students needed a pretty serious refresher on vector calculus anyway, so it is tempting to give it a fresh, modern face when doing so. My big concern is that the semester will melt away as we relearn so much calculus, and we will never see any geometry.

Figure of overlapping coordinate patches

I am really pleased with the quality of the exposition. What I read was well written, contained many good figures, lots of well-chosen and well-explained examples, and had many good exercises accompanying the development. The depth and care of the exposition is wonderful and make this an excellent choice for someone learning the subject on their own. And the fact that O’Neill redevelops multivariable calculus in the way he needs it means that the set of prerequisites really is some passing acquaintance with standard first and second year American college mathematics courses.

But there is a down side to the choice of moving frames. The chapter I read introducing surfaces is 70 pages long, and it has no geometry in it. At the end the reader really understands what a mathematician means by the word “surface,” and how one might hope to do calculus on one. But there is no discussion of anything resembling geometry. I think I saw the coefficients of Gauss’ first fundamental form in an exercise.

I wanted to see how the geometry would work out, so I read some of chapter five, Shape Operators, too. This has a fairly standard outlook, and there are no differential forms to be found.

### Verdict

This book is a decent choice for an undergraduate introductory course. It will feel like exactly the right choice if you want to use Cartan’s approach through exterior algebra of forms. If not, you will find yourself taking a long detour to get to the geometry. I think O’Neill has done a very good job of gauging the right level for a typical undergraduate in the United States. There is lots of support for the newcomer in the form of excellent examples and exercises.

### The MathSciNet Review

This is the Mathematical Reviews item on the first edition. (The update for the second review is contentless.)