Here’s an idea I had for a book. Maybe someone has already written it. If you know of such a book, please let me know.

Differential geometry has a huge ratio of definitions to theorems. It seems like you do nothing but study definitions for a semester or two in preparation for proving something later. **It’s easy to lose sight of the geometry**. I’d like to see a book that is a concrete complement more typical abstract books.

My suggestion is for someone to write a book that goes through a standard differential geometry book, like Spivak’s, and compute everything for a small number of example manifolds: at least a **sphere** and an **ellipsoid**, maybe a **torus**. The book would first go through everything on a sphere where things are simplest, then generalize to an ellipsoid. There would be a lot of applications to geodesy: to first approximation the earth is a sphere, to second approximation it is an ellipsoid.

Sometimes a calculation, such as arc length, is very simple on a sphere. It can be done just using trig. Then the analogous calculation on an ellipsoid is much harder. It is complicated enough to illustrate the machinery of differential geometry. However, we know the answers shouldn’t be much different from those for a sphere, so we have a way to see whether the results are reasonable. This book would not shy away from computational difficulties.

I imagine this book would have lots of illustrations. It might even come with physical models, such as a globe with an exaggerated equatorial bulge. **The idea is to be as tangible as other books are abstract**.

I don’t plan to write this book, at least not any time soon. Maybe if my consulting goes well I would have the time to work on it in the future, but **now is not the time for me to write a book**. In the mean time, if someone wants to scoop my idea, please do!

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Here are a couple other book ideas I’ve blogged about: R: The Good Parts and a rigorous elementary statistics text.

And here are some posts on geodesy: What is the shape of the earth? and Latitude doesn’t exactly mean what I thought.

And finally a few posts on spherical geometry: Napier’s mnemonic, The Sydney Opera House, and Mercator projection.

…Maybe throw in some example with negative curvature example as well?

That could be an exercise for the reader.

My first exposure to geometry was this book: http://www.pearsonhighered.com/educator/product/Differential-Geometry-of-Curves-and-Surfaces/9780132125895.page

It’s not exactly what you’re asking for, but it’s definitely more concrete with more computation (and more of a 4th year undergrad than 1st year grad school book) than Spivak’s books or Do Carmo’s 1st year grad school book. Very enjoyable, too.

I can’t remember how much computation Spivak does in the exercises. For better or worse, I think it’s pretty standard to pull almost all the concrete examples there.

Do you have an opinion about Burke’s “Applied Differential Geometry”? Some reviews I read hint that it’s quite different than other books — more oriented towards building intuition, with lots of physical examples and illustrations (my knowledge of DG is close to zero, still figuring out how to learn this stuff).

Have a look at:

http://www.amazon.co.uk/Riemannian-Geometry-Beginners-Second-Edition/dp/1568810733

John, have you seen Sussman and Wisdom’s totally excellent “functional differential geometry”? http://groups.csail.mit.edu/mac/users/gjs/6946/calculus-indexed.pdf It’s not what you’re asking for, but it’s a very exciting take on the subject (as a computer scientist)

A tangible and fairly concrete book on Earth geodesy is

Map Projections: A Working Manual — 1987, Snyder, John P.

now apparently only only available as a PDF which is a shame as the original print version had some great large format fold out diagrams of various projection systems.

The historical pre-GPS working manuals for geodesy were a right dogs breakfast for the globally nimble surveyor as there were several hundred archaic variations of 5 or 6 projection systems and comprehensive almanacs of conversion parameters could do serious damage to feet if dropped.

Synder’s book covers the mechanics of calculation and the formulas, approaches and common approximations with various worked examples. It doesn’t attempt to all the variations between all the systems ever used (the only “book” I’ve seen that attempted that was the compiled private notes of another surveyor).

This has little of the theory of differential geometry (as some grasp of that is assumed) but a major league professional focus on the mechanics of geodesy; Snyder was the creator of the space-oblique mercator projection after all (and had a fair few other feathers in his hat).

http://en.wikipedia.org/wiki/John_P._Snyder

http://pubs.er.usgs.gov/publication/pp1395

Buy, beg, borrow, or steal a print copy of this one if maps are your thing.

Eric: I skimmed that book today when someone told me about it. It’s superb at making calculations concrete, so concrete a computer can do them! But it doesn’t do much for geometric intuition.

I remember stumbling across a differential geometry book like that many years ago, in the library. It was full of pictures of curves and surfaces, with tangent planes etc. all nicely illustrated. Some computing software like Mathematica was used to help with numerical calculations, and to make the plots. Hmm, I wonder if I can find it on Amazon…

Maybe it’s this one?: Modern Differential Geometry of Curves and Surfaces with Mathematica, by Abbena, Gray and Salamon. It’s already in a third edition.

The reason why the ratio of definitions to theorems is large in differential geometry is that many theorems of classical vector analysis, tensor calculus, and integration are “built-in” into the definitions.

The amount of work you need to put in to understand the definitions is almost the same you would put in to understand the theorems of the classical theories.

However, the return of investment in studying the differential geometry is far greater. Once you have a firm grasp of the definitions, the more advanced theorems that are unbearable in the classical theories are suddenly at your reach to understand.

I agree with your reasons for why the machinery of differential geometry is set up the way it is. It’s optimized for the researcher working in the area. However, it makes it harder to get started in the field. I think a book like I propose might be a good companion to a standard differential geometry course as a way to remind students that they really are studying

geometry.There’s a similar problem in probability (and other areas of math too for that matter). In the middle of a measure-theoretic probability class, a student might ask how it all relates back to an intuitive idea of probability. But at least in to my mind, the gap between formalism and intuition is easier to bridge in probability than in geometry.

I agree with Troy McConaghy. The book Modern Differential Geometry of Curves and Surfaces with Mathematica makes the abstract concepts concrete. It is interesting to hear that the book is now in its Third Edition because the original author, Alfred Gray, dies shortly after the first edition (which I own) was published.

Adams and Guillemin is an undergraduate textbook designed to address the issue of measure theory for probability.

Like David Chudzicki, you’re supposed to learn the geometry of curves and surfaces in 3-space before moving on to Riemannian geometry. This is the geometry that Gauss inherited and developed as a surveyor. Such a course culminates with the Theorem Egregium, thus motivating intrinsic geometry. Do Carmo wrote texts on both subjects, so may say something explicit about the transition. I also recommend Struik’s book, available from Dover for $10.

Your post reminds me that the closest I ever come to feeling like I understanding a (pure math) subject is when I can visualize it in such a way that (1) The truth of theorem are obvious, and (2) The way to prove a theorem follows straight from the picture. This certainly was true when I was learning topological manifolds, smooth manifolds, and differential geometry. Though I guess it’s true in most technical subjects–you really start to feel like you grasp the material when you can come up with both simple counter-examples as well as simple examples that encapsulate the general argument.

In any case, two of my favorite abstract math books are Visual Group Theory (by Carter) and Visual Complex Analysis by Needham. I like them precisely because they emphasize visual, easy-to-grasp examples.

I’d like to read Carter’s book. I have Needham’s book and it’s great.

I really like this one, which Michael Betancourt (a physicist) recommended:

Baez and Muniain. 1994.

Gauge Fields, Knots, and Gravity.The first few chapters provide a really beautiful presentation of differential manifolds and related concepts in the modern style (i.e., more vectors as vectors algebra than a ton of index fiddling). It’s more concrete than the presentations I’ve seen from mathematicians and somehow manages to make it all seem simple and natural. There are even good get-familiar-with-the-notation-and-prove-a-few-extensions exercises. It’s one of the best math books I’ve ever read and I get to learn some physics along the way. (My real motivation is understanding Riemann manifold Hamiltonian Monte Carlo.)

Bob: Sounds like a good book. I’ll add that to my list.