more of “The Poincare Conjecture”

 
It seems a little unfair that I listed Weeks’ “The Shape of Space” in my bibliography some time before I listed O’Shea’s “the Poincare conjecture”. Oh, “the shape of space” is a fine book, and I’m very glad I got it, but it was O’Shea who got me started – restarted – on all this.
I noticed this omission when I was editing the bibliography. I omitted it just because I wanted to say more about the O’Shea book but hadn’t figured out exactly what to say.

O’Shea’s “the Poincare conjecture” is, by and large, a pleasant mix of biography and history, and it mentions some fascinating mathematics, but – quite appropriately – it barely scratches the surface. I want more. I hope, at the very least, that some of its readers end up saying things like:
“He makes (Euclid’s) geometry sound a lot more interesting than it was in high school.”
“He talked about the non-Euclidean mathematicians rather than non-Euclidean mathematics. Where do I find the mathematics?”
“What do you mean, there are an infinite number of ways to do calculus in 4D space?”
There are 3 things that stand out for me in “the Poincare conjecture”. 
The easiest to mention is its reference to V. I. Arnold’s “On Teaching Mathematics”, which can be found in HTML and PDF at
and
respectively. It’s a tirade – by an outstanding mathematician – about abstract and formal mathematics divorced from its roots. My favorite story from it is of the French schoolchild who is asked, “what is 2+3?” He answers, “it’s 3+2, because addition is commutative.”
True, but some of us were hoping to hear “5”. Some of us think “5” is the essential answer.
He also says he taught group theory to schoolchildren in Russia, and the notes of the class are available in English. That book just arrived this afternoon. I haven’t had a chance to look at it, but i’m eager to see what he did.
The next easiest to mention is O’Shea’s “further reading”. Here is where I found Weeks’ “the shape of space”, which tries to do what Arnold says: play with surfaces, play with ways to understand 3D manifolds, do not just go for abstraction. Weeks was a student of Bill Thurston, who apparently not only had an awesome geometric imagination, but also encouraged intuition in others. Some notes and a book of Thurston’s are listed, along with a single-volume exposition of Perelman’s proof of the Poincare conjecture, “ricci flow and the Poincare conjecture” by Morgan & Tian. 
The Thurston book is on its way to me. The “Ricci flow” has already arrived. 
Yes, I bought it too. No, I can’t read it. But until I can, I’ll know that there’s more for me to understand in my intermediate books.
The most outstanding thing in “the Poincare conjecture” was his statement that “… Poincare & Klein’s work implied that any surface could be given a geometry in which it had a constant curvature….”
That blew me away. I’m embarrassed, but not as much as I might be: none of my books ever phrased it that way. I have books about surfaces of constant curvature, and none of them said that they included, in some sense, all surfaces.
It took me a while to find out what O’Shea was talking about. I looked through old familiar – obviously not that familiar – texts, but Weeks gave it to me. He phrased it as “every surface can be given some homogeneous geometry.” And he went on to explain how.
The embarrassing part is that it’s not only a named theorem, but it’s one of those “the major theorem of this book” theorems. 
Its name is Gauss-Bonnet.
What it says is that an apparently geometric quantity (the total curvature, i.e. the integral of the curvature at each point) is equal to a topological invariant (the Euler characteristic or Euler number, vis. the number of vertices plus faces minus edges).
Ok. But that doesn’t say anything about changing the geometry.
But here’s what O’Shea and Weeks mean. Take a donut. More precisely, take the surface of a donut. Or take an inner tube. These are 2D surfaces if we imagine them of zero thickness – like all the lines we drew with straight-edge or compass in high school, that really had width but we imaged they didn’t. 
The donut sits in 3-space, and it acquires an induced geometry from that 3-space: we can compute the lengths of paths on this donut, as we can compute lengths of paths on the surface of the earth, using Euclidean geometry but staying on the surface. We’re not allowed to tunnel from here to china; must go by land and sea. Ok, by plane, too.
More to the point, we can compute the curvature at every point on the surface of the donut; and then compute the total curvature by integrating. 
We get zero. The curvature isn’t zero everywhere: it’s positive on the outer parts, and negative on the inner parts, but the total is zero.
Now we do something topological. We cut the donut and unbend it so that it becomes a cylinder. Then we cut the cylinder the long way and – voila’ – we have a flat piece of paper.
The total curvature of the cylinder is zero, like the donut, but more importantly, the curvature of the cylinder or of the piece of paper is zero everywhere.
It was that unbending, stretching the inner part of the donut and compressing the outer, that changed the curvature to zero everywhere.
We have given the donut a homogeneous geometry, one in which it has constant curvature. The key restriction is that we have preserved the Euler number, so each of the total curvatures remains zero. in particular the constant curvature is zero. Topological transformations won’t let us change that.
Want to learn more about this? For intuition and understanding, read Weeks. For the actual mathematics, I’d go with O’Neill’s “elementary differential geometry”, 2nd ed. If you want to see even more about it, Bloch’s “a first course in geometric topology and differential geometry” will talk about topological, polyhedral, and smooth structures on surfaces. (All of these are undergraduate texts, and the Weeks book is probably accessible to high school readers.)
Yes, the three structures coincide for surfaces. But it was only a few years ago that I learned “topologists study three types of manifolds – topological or continuous… piecewise linear… differentiable….” it was news to me. (That’s the opening sentence of the introduction to another one of those books I can’t read yet, freed & uhlenbeck’s “instantons and four-manifolds”.)
I bought the Bloch book because it would show me all three structures between two covers. I have just started actually reading it. If you’ll pardon my saying so, “Yummy.”
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