It is time to return to the book reviews! Our next book is

*Elementary Differential Geometry, 2nd Ed*by Andrew Pressley. This is a pretty recent text. The first edition is from 2002, with the update published in 2010. The book has an attractive price point from Springer, and you can get it from Amazon.com for even cheaper.

Pressley’s desired approach is to make the subject as accessible as possible. In the preface, he writes:

Thus, for virtually all of the book, the only prerequisites are a good working knowledge of Calculus (including partial differentiation), Vectors and Linear Algebra (including matrices and determinants).

The tone of the writing bears this out, as does the author’s care to explain basic material. This text is definitely aimed at the modern student, and it conforms to the standard expectations for what a recent textbook on an advanced subject should look like.

### The Chapter Headings

Pressley has organized the material as follows:

- 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

(As you can see, Pressley doesn’t use a serial comma. We are already at odds.) There are also three appendices, enumerated computer-science-style:

- A0. Inner product spaces and self-adjoint linear maps
- A1. Isometries of Euclidean Spaces
- A2. Möbius Transformations

Clearly, these are chosen to support some of the prerequisite material for describing curvature, understanding congruence, and dealing with hyperbolic geometry.

Keeping to the plan, I read chapter 4 *Surfaces in three dimensions*.

### Notable Features

As you can see from the chapter headings, Pressley wants to lay out the basics very carefully. There are whole chapters devoted to the basic process of defining curves (chapter 1) and defining smooth surfaces (chapter 4). The writing is direct but relatively friendly. You don’t get the “marble temple of mysteries” feel that some advanced textbooks can fall into.

There are over 200 exercises. (I am quoting the preface for this. I didn’t count.) There is a selection of hints for about 75 of them near the end of the text. (I counted that myself.) More importantly, there is also a 60 page section at the end of the book with fairly complete solutions to all of the exercises. Some seem a bit terse, but they do all seem to be addressed. This makes the text a really good choice for individual study. A person with the self-discipline to make an honest effort at each exercise would be glad of this feature.

Some of the exercises I saw were pretty hard. That is, an undergraduate with the background listed in the preface would likely struggle mightily with some of them, though for the initiated they would be not difficult.

Exercise 4.1.4: Show that a unit cylinder can be covered by a single surface patch, but that the unit sphere cannot. (The second part requires some point set topology.)

That parenthetical remark is in the text.

Exercise 4.1.5: Show that every open subset of a surface is a surface.

A person with a fair amount of experience can dispatch those, but a student making the transition from calculus and linear algebra is likely to spin their wheels a long time. Those exercises just go straight to the “fiddly bits.”

We have the now standard, *definition-theorem-proof* style of exposition, and plenty of computer generated diagrams. Everything is labeled in one consecutive sequence in *section.subsection.item* style.

The examples are the ones you would expect, or, rather, they are the ones I expect. I recognize that I am not typical by a long way, having been trained in the subject formally, and now undertaking my seventh book review of this material. It is nice that the examples reoccur when new concepts arise.

Pressley has included more material than can be reasonably discussed in a one semester course. In particular, it looks as though you can, and should, pick a path through the text to *one* of the final three chapters. This gives the book a little flexibility, and it leaves more material to whet the appetites of ambitious students.

### A Complaint, only partly about the book

I have a big complaint to make, but I don’t think it is altogether fair to Prof Pressley. If anything, he is merely exemplifying a trend in mathematical exposition that has important uses, but which I have come to question about the construction of learning materials.

It is standard practice in mathematics to present things axiomatically. This has been going for a long time: I teach out of Euclid’s *Elements*, and it happens there. You get formal definitions and axioms first, theorems later, examples and discussion sometimes.

We don’t have to have the conversation about why this is so. We can all agree that it is generally a good thing to have the axiomatic structure in mathematical work.

I can’t really fault Pressley for doing things this way. This is

.How Mathematics Is Supposed To Be Written

(Did you hear the echo? I heard an echo.) If you want to prove theorems you actually suspect are true, this is how it goes.

But I don’t necessarily think it is the right thing for curricular materials. To really appreciate the nuances of mathematical definitions, one needs lots of examples. Even more, to even really feel a need for all of these crazy words (and find a will for keeping them straight), you have to have lots of funny friends that you can classify and organize with those words.

The upshot is that I believe that chapters 4 and 5 are in the wrong order.

Here is how the standard axiomatic approach leads to trouble for the potential reader in this text. Chapter 4 is about conveying the idea of a surface, and getting a definition down. But the first subsection *4.1 What is a surface?*, opens with a page of definition-making in which we see the following terms:

- open subset ( definition),
- open ball
- open interval
- open disc (Pressley is from the UK, so we get British English spellings)
- functions continuous at a point ( definition)
- functions continuous in the large
- homeomorphism
- homeomorphic spaces

Then, and only then do we meet the first official definition, Definition 4.1.1, which gives the formal atlas of patches definition of a surface, essentially the notion of a topological 2-manifold. To Pressley’s credit, he doesn’t actually use the word manifold.

All of that is over in about a page of text, and then basic examples start. It seems to me that the preceding would be very hard to comprehend for anyone who hasn’t already mastered some point-set topology, or at least been exposed the ideas of metric spaces. The young, aspiring geometer just won’t have a sense of what those words are for, and why we have chosen to use all of them.

Before the chapter is out, we are introduced to regular surfaces, diffeomorphisms, the derivative of a function from one surface to another, orientability, the notion of a maximal atlas, the tangent plane as the set of all tangent vectors to curves through the given point which lie in the surface, and other things in this neighborhood of mathematics.

In a weird way, I think that all of this axiomatically clean development puts the cart before the horse for newcomers.

So, though Pressley has made an admirable attempt to be clear and helpful, I find I am dissatisfied. Basically this comes across as an extraordinarily friendly introduction, written so as to be useful more as a reference than as a learning tool. Again, not exactly the author’s fault…

### Verdict

A solid, mathematician’s introduction to the subject. Perhaps misses its mark of being truly accessible to the pre-analysis crowd. Any real flaws here are flaws in almost any advanced mathematics textbook. Instructors should make themselves aware of the book, as it could be a reasonable choice for an instructor who puts in the effort to help students through the transition.

This is a decent selection for someone with more advanced training who wishes to learn the subject by studying independently.

I get the sense that you’re bumping up against one of the fundamental problems in mathematics education: the way our learning of mathematics is best structured does not always match the way the mathematics itself is structured. Unfortunately, figuring out the learning part can be quite difficult, and perhaps the only place where we can claim to broadly understand it well is for basic arithmetic.

I know some mathematicians lament the “demise” of a strict axiomatic approach in high school geometry. I think that understanding an axiomatic system is important, and that’s why I would typically spend several days teaching directly from Book I of Euclid’s Elements. But my goal for doing so wasn’t for students to really understand the mathematics of the propositions being proven — I only expected them to gain an appreciation for how an axiomatic system is developed. If my goal was for students to know and understand how to use the Pythagorean Theorem, there were certainly better options than simply marching my way to Proposition 47. I think you’re looking for those kinds of options for topics in differential geometry, which I’m sure will take some creativity and risk-taking to develop and test. Good luck!

Well, risk-taking is my business, I guess. I am not certain that the students appreciate it all the time.