So far, all of the books I have reviewed have been old ones. That is, the “newest” book was do Carmo, from the 1970’s. Now it is time for a recent book, *Differential Geometry of Curves and Surfaces* by Thomas Banchoff and Stephen Lovett (2010). The book is published by A K Peters, Ltd. which is an imprint of CRC Press. The text is also available from Amazon.com. The Amazon.com page also gives you the option to buy electronic versions!

Being a more recent book, several features of more modern textbooks are readily evident: the now-standard *Definition-Theorem-Proof* style, many examples and exercises, and some Java applets available from a publisher-hosted web site. That same page has a link to the list of errata.

This book appears to be written for a current undergraduate audience, and it comes the closest to meeting my idea of the right set of prerequisites: multivariable calculus and linear algebra, and maybe some rudiments of ordinary differential equations. If you are choosing a new text, you really should have a look at this one.

### The Chapter Headings

The text is organized in the following chapters.

- Plane Curves: Local Properties
- Plane Curves: Global Properties
- Curves in Space: Local Properties
- Curves in Space: Global Properties
- Regular Surfaces
- The First and Second Fundamental Form
- The Fundamental Equations of Surfaces
- Curves on Surfaces

Since the plan is to read whichever part of the book introduces surfaces, I read chapter 5. I am starting to regret this a little bit, because it means that I am not necessarily reading any *geometry*. I am reading about the underlying differential topology that goes into defining a surface. But I am going to stick with the plan, because this is a place where students have a lot of difficulty. It is important that this material be covered well, or the geometry won’t happen anyway.

### Points of Note

The authors have gauged the right level of pre-requisite material, in my opinion. A student who has successfully completed the calculus sequence and a linear algebra class has a chance to read and learn from this book.

The selection of material is just right for a one semester course. For each phase of discussion (plane curves, space curves, surfaces), the basic local theory is discussed and one or two important global results are developed. I saw the Four Vertex Theorem for plane curves, the Fary-Milnor theorem for space curves, and geodesic coordinates and the Gauss-Bonnet theorem for surfaces. A brisk semester can get through the whole book.

The writing is clear, and there are many examples and ample exercises of varying levels.

### The Java Applets

The most interesting feature of the book is the set of Java applets. For chapter 5, there were 14 different spots labelled with a little laptop picture indicating the existence of a corresponding applet. Most of these are examples.

The applets are well-done. That is, they are visually attractive, and they allow the user lots of freedom to play around. Just about anything you can hope to change is allowed as a parameter, either displayed explicitly as an input, or hidden under one of the menus. I found them easy to use and customize. I would like to think that students would also find them intuitive, but I have learned that I am different from the typical student in so many ways, that I hesitate to say this firmly.

I especially like the choice to separate each applet into three windows. There is one window with all of the explicit controls: buttons, input boxes, parameter sliders, etc. The other two windows are for display. This is the best part, there is a window showing the domain of the relevant patch in the plane, with the vertical and horizontal coordinate lines clearly outlined in different colors, and there is a window displaying the surface as rendered in a three dimensional view, with the images of those coordinate lines in the same colors. As you move your point of interest around, all of these things change in sync. I can’t think of a better way to reinforce the idea that a surface is really built as a parametrized object, and it is the function which helps us do geometry. This is a stumbling point for most students, and the applets are going to make a positive difference on that score.

On the downside, the applets required Java 7. This meant that I had to update my Java and switch to Firefox, as my usual browser (Google Chrome) and this version of Java are incompatible. In fact, I have a newish MacBook Pro, and so I also got lots of security warnings when I did a quick internet search. Apparently Apple thinks little of Java these days. This also means that I couldn’t try the applets on an iPad.

### The Verdict

This is a nice book. I am pleased with the parts that I have read. I will definitely read more of it. If you will teach an elementary differential geometry course, you should be aware of this book and give it serious consideration as a text or a reference.

### The MathSciNet Review

Mathematical Reviews has a short review of the object, which is mostly a description of the contents. The real review is encapsulated by only the last sentence:

In summary, the authors succeeded in making this modern view of differential geometry of curves and surfaces an approachable subject for advanced undergraduates.

I agree.

would you consider Banchoff and Lovett’s Differential Geometry of Curves and Surfaces a good introdcution for self-learners?

I think it is possible to use the text to learn the subject independently.

Be sure to dig in with the applets.

The important thing is to build your intuition and your technical fluency at the same time. This is challenging in Differential Geometry. Personally, I learned the intuition pretty fast, but technical fluency was slow. I had to do lots of exercises, fail at them, get help, and do them again. For this, I like Pressley a bit better.

But it is probably a good idea to use more than one reference, in any case.