★★★★★ Visual Differential Geometry and Forms: A Mathematical Drama in Five Acts, by Tristan Needham

Putting the geometry back in Differential Geometry

Visual Differential Geometry and Forms: A Mathematical Drama in Five Acts

Tristan Needham

Putting the geometry back in Differential Geometry

If you’re a mathematician, you are repeatedly reminded that there are vast reaches of mathematics, important and beautiful, of which you personally know almost nothing. (I am, at least.) For me one of those areas is differential geometry. Oversimplifying a bit, differential geometry is the study of curved spaces. It’s important to physicists mainly because, according to Einstein (and a hundred years of experimental results) gravity is caused by the curvature of space and time. I have studied the math of gravity, so I am not entirely ignorant of differential geometry. I didn’t really understand it, though. It was just a bunch of formulas that didn’t really make much sense.

Thus, when I recently discovered that Tristan Needham had published a textbook of differential geometry (VDG below), I was intrigued. I was intrigued because many years ago I read Needham‘s other book, Visual Complex Analysis (VCA in what follows), and it was great!

Needham begins his preface with this quote,

Algebra is the offer made by the devil to the mathematician. The devil says: “I will give you this powerful machine, and it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvellous machine.”... the danger to our soul is there, because when you pass over into algebraic calculation, essentially you stop thinking: you stop thinking geometrically, you stop thinking about the meaning.
-- Sir Michael Atiyah

That felt exactly like my experience with gravity.

Needham is a man who loves, loves, loves geometry, more than almost anything. In his Acknowledgements he claims to love his wife Mary and his twin daughters more, but it is obvious that the competition is close.

Now for a confession. The right way to read a math textbook is to work through it carefully, making sure you understand the proofs and working the exercises. Reading a math text without working the exercises is about as effective as visiting a gym and reading the instructions for the machines and the free weights, without actually using them. Working the exercises takes a long time, and I didn’t want to spend so much of my life on this book. Thus I decided to read it like a novel. Needham‘s Visual Complex Analysis can be read that way (it’s the pictures that do it), so I had hopes that this would still be worthwhile. Those hopes were mostly justified.

Needham‘s account of differential geometry is divided, like a play, into five acts. The first three acts present what I will call classical (that is, pre-Riemann) differential geometry, which is mostly about the geometry of two- dimensional spaces. Examples would be the surface of the Earth. The Earth is a three-dimensional object, but its surface is pretty close to a a two-dimensional space. It’s a curved two-dimensional space. Acts 1-3 mostly explore the geometry of two-dimensional spaces. This was almost entirely new to me, and it was the greatest win of the “read it like a novel” gamble. I now understand classical differential geometry far better than I did before reading

Act IV extends the discussion to spaces of more than two dimensions. This was Bernhard Riemann‘s great contribution to differential geometry, which became the mathematical basis of Einstein’s theory of gravity. No one knows how Riemann did it. In spaces of more than two dimensions, the curvature at each point in space is a complicated object now called the “Riemann tensor.” Every modern treatment of the Riemann tensor (including the ones I had studied previously) depends on an idea called “parallel transport” that would not be invented until 1917, long after Riemann‘s death in 1866, and even after Einstein‘s publication of his theory of gravity. Needham‘s Act IV was not a noticeable improvement on the previous presentations of general relativity that had left me unsatisfied in the past. It felt as if he had given in to the Devil’s bargain described in the quote -- it was mostly algebra, without much of Needham‘s visual geometric intuition. There probably is a better way to do this, because Riemann did it. But Riemann is possibly the best mathematician of all time, so I can’t really fault Needham for not finding it.

This is especially true because he redeems himself in Act V, “Forms,” where he presents Elie Cartan‘s theory of forms and uses it for a visual, geometric presentation of differential geometry in more than two dimensions. Cartan‘s forms were almost entire new math to me. Act V was the greatest failure of my “read it like a novel” strategy. I benefitted, but I would have learned a lot more if I had worked the exercises. Perhaps I will, one day.

I’m not sure if I have conveyed how clearly Needham‘s voice sounds throughout VDG. He’s a Star Trek fan¹ who is deeply moved by mathematical beauty².

Visual Differential Geometry on Amazon

Goodreads review

1. The index entry for “Star Trek” speaks for itself
Star Trek
Captain Kirk, 191, 482
Dr. McCoy, 38
forms proof of (“Star Trek phaser”) metric curvature formula, 452
geometric proof of (“Star Trek phaser”) metric curvature formula, 266
Mr. Spock, 191, 476, 482
NCC-1701, 435
(“Star Trek phaser”) metric curvature formula, 38
The City on the Edge of Forever, 38
The Doomsday Machine, 191

2. He describes the flash of insight that led him to a geometrical proof of one of the most important theorems of classical differential geometry in a footnote, “the startling, unanticipated flash of clarity—in the Sierra mountains, surrounded by pristine snow—was one of the happiest moments of my life.”