Thursday, 31 January 2013

Today I Learned: On De-Klein-ing Lie's - Part 2

Update: Part 0 Part 1.

Roughly a year ago, I waltzed through the streets of Munich.

Everything was wonderful - I'd recently befriended an American couple and a African+German couple (they all very kindly let me crash at their places for weeks) and life was a whimsical concoction of Japanese conversations with strangers (I'm not Japanese), Bible studies, scaling European architecture (in town and on the alps), office meetings with my supervisor at the Ludwig Maximilian University (he was technically holidaying) and asking God to provide for all of my accommodations (I'm currently fairly well-off, but I asked it for good reason, and He did provide for a whole month - including a free week at a hotel in Istanbul, so I'd better hold up my end of the bargain). Even the weather was wonderful: cold, lippy, but not so chilling as to be penetrating.

The only thing that really bothered me was...why did they love Bayesian Statistics so much? I mean, the signs were everywhere. Above libraries, on opera posters, on random landmarks were scribed these...these impenetrable and insufferable letters: "Bayerische Staats". They haunted me. Statistics had made her mark upon the capital of Bavaria.

Yes. Bavaria. I only realised (much to my humiliation) after I'd landed back in Melbourne that I'd mistranslated Bayerische Staats, that is: the Bavarian State.

It was a good trip, I even managed to miss my flight back to Australia because I was doing some maths with my supervisor and scored having my first Thanksgiving turkey - with the American couple I mentioned earlier. Life (*) is awesome.

Pity that I hadn't also visited the second largest city in Bavaria - neighbour to a little university/Siemens town called Erlangen.

A while back (to all AMSI peeps who're still reading this, after that meandering introduction, yes, I know that your summer school is nearly over and your exams are coming up - hey, I'm a slow writer), I decided to write up a few things that I've learned from reading one of Klein's lectures on Lie's line-sphere correspondence. This is part 2: the Erlangen program.

A slightly modern understanding of the Erlangen program is that it is the study of the invariants of a group \(G\) acting transitively on a homogeneous space \(X\). There are key words here: groups, transitive, spaces and invariants. Let's consider an example.

Example 1.

The group \(SO(n)\) made up of orthogonal matrices with determinant \(1\) may be thought of as matrices whose column (and rows) constitute an orthonormal basis for \(\mathbb{R}^n\). This means that:
  • any vector gets mapped to a vector of the same length;
  • angles between vectors are preserved.
The only transformations of \(\mathbb{R}^n\) which can do this are rotations and reflections - we discount the latter option in this case because the determinant of a matrix in \(SO(n)\) is \(1\) as, opposed to \(-1\).

Now, because orthogonal matrices must preserve vector lengths, they cannot act transitively on all of \(\mathbb{R}^n\), so our space needs to be something smaller - like the unit sphere (any radius will do):\[\mathbb{S}^{n-1}:=\{(x_1,\ldots,x_{n})\,:\,x_1^2+\ldots+x_{n}^2=1\}.\]

The transitivity of the action of \(SO(n)\) on \(\mathbb{S}^{n-1}\) means that for any two points \(x,y\in \mathbb{S}^{n-1}\) there are a whole bunch of matrices taking \(x\) to \(y\), and either by post- or pre-multiplication by a matrix, it's clear that this set of matrices is in bijection with the group of matrices fixing \(x\) (or fixing \(y\)). Take \[x=(0,0,\ldots,0,1)\] and we see that this group of point-fixing matrices needs to be \(x\) in the bottom row and \(x^{t}\) in the last column, so the group of point-fixing matrices is precisely \(SO(n-1)\). Which of course means that: \[\mathbb{S}^{n-1}=SO(n)/SO(n-1).\] And this is what we mean by a homogeneous space: a group quotiented by a sufficiently nice (possibly non-normal) subgroup.

So, we've got our group, we've got our space - what are these invariants?

Well, we know that \(SO(n)\) is distance preserving on \(\mathbb{R}^n\), and knowing this it's not too hard to show that orthogonal matrices also preserve distances on \(\mathbb{S}^{n-1}\). So there you go: distances are preserved. In addition, the angle between two intersecting curves on \(\mathbb{S}^{n-1}\) are also preserved - you can show this by thinking about the tangent lines of these two curves at the relevant point of intersection (an exercise!). So, distances are preserved, angles are preserved, surely this means that spherical polygons must be preserved (insert higher dimensional generalisations of polygons where appropriate).

In fact, we're seeing that these invariants are pretty much what we'd expect to care about as geometers: spherical lengths, angles, spherical shapes, spherical volumes...etc.

In fact, it's as if:

the Erlangen program is an attempt to unify geometry in a cohesive language.

Let's list a few more examples:

Example 2.

Group: \(O(n)\) - the orthonormal matrices.
Space: \(\mathbb{S}^{n-1}\)
Invariants: spherical lengths, angles, spherical shapes, spherical volumes...etc.

Wait, isn't this the same example? Well, almost. The only difference being that \(O(n)\) includes not just rotations but reflection matrices too. And so there is one invariant that is retained for \(SO(n)\) that isn't for \(O(n)\): orientation (mirror-image-ness). So in our first example, we were able to distinguish a triangle on \(\mathbb{S}^2\) and its mirror image, whereas the two would be regarded as being identical in this second example.

Example 3.

Group: the group of linear functions look-a-likes on \(\mathbb{R^n}\) of the form: \[x\mapsto Ax+b,\]
where \(A\in O(n)\) is an orthogonal matrix and  \(b\in\mathbb{R^n}\) is just another vector.
Space: \(\mathbb{R}^n\).
Invariants: lengths, angles, shapes, volumes...etc.

This might look unfamiliar, but it's really just Euclidean geometry in disguise. Our space is \(\mathbb{R}\) and our symmetries are any combination of translations and rotations and reflections - precisely the symmetries that we'd think of in primary school.

Example 4.

Group: the special linear group \(SL_2(\mathbb{R})\) of \(2\times2\) matrices of determinant \(1\).
Space: the hyperbolic upper half plane \(\mathbb{H}\) given by \[\mathbb{H}:=\{z=x+yi\in\mathbb{C}\,:\,\mathrm{Im}(z)=y>0\},\]
and the action of the group is given by:\begin{align}\left(\begin{array}{c c}a & b \\ c & d \end{array}\right)\cdot z:=\frac{az+b}{cz+d}.\end{align}
Invariants: angles, orientation, hyperbolic length, hyperbolic polygons, hyperbolic area...etc.

It's sometimes said that polygons in in spherical, Euclidean and hyperbolic geometry are respectively fat, boring and thin (okay, I sorta lied: I haven't heard anyone other than high school kids call Euclidean triangles "boring"). As in, the sides of spherical triangles are curved slightly outwards, the sides of a Euclidean triangle are straight and the sides of a hyperbolic triangle are curved slightly inwards. In fact, this simple phenomenon is a manifestation of a deeper invariant of these geometries called the Gaussian curvature. In particular, the curvature of a sphere is positive, for the plane it's zero and for the hyperbolic plane it's negative, and you can think of curvature either in terms of how triangles curve or how much "space" you squeeze in at each "point".

Also, it's worth pointing out that the fact that the hyperbolic plane is a homogeneous space does make it extremely rigid. I'm just mentioning that as an alternate handwavy explanation of the rigidity of hyperbolic spaces to any individuals who might have asked me about this in the past.

It's also worth pointing out that I've provided a presentation of the hyperbolic plane that does not generalise very well to general hyperbolic spaces (and instead generalises more nicely to the Siegel upper half space). Go check out the Wikipedia article for homogeneous spaces to see how to do it for hyperbolic spaces.

Recently, I stumbled across this discussion on MathOverflow about the Erlangen program and what precisely it states. And I think that one of take home message that I got was that Erlangen program is that curious combination of beautiful logic and deep intuition - in parallel with Klein's self-assessment as a mathematician (perhaps not so colourfully phrased). It's like Monet's impressionism: colour, fire, light, captured by masterful brush strokes. Its triumph is that of intuition - we see once and once again the her indelible fingerprints in geometry. In Riemannian geometry, she is present at the grand level by uniformization and local models and present at the microscopic level via the Levi-Civita connection. Which in turn leads us to Cartan geometries and Cartan connections. In symplectic geometry, Lie's contact-transformations seem also to be grounded in this philosophy. The details are somewhat foggy to me, perhaps like a Seurat, but the forms are there. Klein, more than just the guy who came up with the Klein bottle.

*: Secretly, I wanted to write that God is awesome, but I chickened out. =P

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