Quaternions and associated manifolds

[mnb96]

Hello,
it is known that pure-quaternions (scalar part equal to zero) identify the $\mathcal{S}^2$ sphere. Similarly unit-quaternions identify points on the $\mathcal{S}^3$ sphere.

Now let's consider quaternions as elements of the Clifford algebra $\mathcal{C}\ell_{0,2}$ and let's consider a quaternion $\mathit{q} = a+b\mathbf{e}_1+c\mathbf{e}_2+d\mathbf{e}_{12}$.
We now re-write q in the following form:

$$\mathit{q} = (a+d\mathbf{e}_{12}) + \mathbf{e}_1(b - c\mathbf{e}_{12}) = \mathit{z_1} + \mathbf{e}_1 \mathit{z_2}$$

We have esentially expressed a quaternion as an element of $\mathbb{C}^2$.

*** My question is:
if we assume that $z_1$ and $z_2$ are unit complex-numbers of the form $$e^{\mathbf{I} \theta}$$, can we find a manifold associated with this subset of quaternions?

 

[arkajad]

assume[/I] that $z_1$ and $z_2$ are unit complex-numbers of the form $$e^{\mathbf{I} \theta}$$, can we find a manifold associated with this subset of quaternions?

I do not quite understand. You parametrize your manifold by two independent circle parameters, you get the torus $$S^1\times S^1$$. Or you mean something else? Do you want to see how it is embedded in $$S^3$$?

 

[mnb96]

arkajad, thanks a lot!
This may be obvious to you, but honestly I didn't realize I was essentially describing a torus: precisely the Clifford Torus.

I am reading now from external sources that the torus I described is a subset of an $S^3$ sphere of radius $\sqrt{2}$. That is very interesting.

Now, I am planning to define a metric on those quaternions lying in the Torus.
I guess one possibility would be to consider the shortest-arc distance for $S^1$, and sum the two shortest-arcs for $$S^1\times S^1$$, but I have to think more about it, I am not so sure.

 

[Ben Niehoff]

I believe you will find that the torus you are describing is in fact flat.

 

[mnb96]

If I want to define a metric for the points on the Clifford torus, how can I use the fact that the manifold has zero curvature?

 

[arkajad]

You have two angles that define coordinates, say $$\theta,\phi$$. Why not to define your metric as $$ds^2=d\theta^2+d\phi^2$$? It's a very good metric.

 

[Ben Niehoff]

Of course, you can always put a flat metric on a torus. The question is whether the embedding in S^3 is flat (hint: it is).

What you should do, technically speaking, is write down the metric on S^3:

$$ds^2 = R^2 (d\theta^2 + \sin^2 \theta \; d\phi^2 + \cos^2 \theta \; d\psi^2)$$

(Hopf coordinates will probably be easiest). Then compute the pullback of this metric onto your torus.

(Note that you get a flat torus out of the above metric for any constant value of $\theta$, and in particular if $\theta=\pi/4$, this torus is "square".)

 

[mnb96]

Thanks a lot!
so summarizing, I can start from the metric on S^3, and since the Clifford Torus embedded in it is flat, I can compute the pullback of that metric into that torus, and (probably) I should obtain the metric that arkajad wrote(?).

Unfortunately I don't know the concept of pullback. Before I dive into the complicated formalisms of books, could you please mention very briefly the intuition behind the pullback of one metric?

 

[arkajad]

If you would like to see your torus - related to quantum spin 1/2 - I was discussing it and drawing http://arkadiusz.jadczyk.salon24.pl/105036,spin-od-srodka"

 

[arkajad]

Pullback: In your case it is simple. You embed your tours in R^4 with coordinates, say x,y,z,w. S^3 is then described by x^2+y^2+z^2+w^2=1. The metric in R^4 is Euclidean.

ds^2=dx^2+dy^2+dz^2+dw^2

Your embedding is given by the functions x(theta,phi),y(theta,phi),z(theta,phi),w(theta,phi).

Then you calculate

$$dx=\frac{\partial x}{\partial\theta}d\theta+\frac{\partial x}{\partial\phi}d\phi$$

then dy,dy,dz,dw, then ds^2. Or, shorter, you embed your torus in S^3 with coordinates $$\theta,\phi,\psi$$ and take the metric from S^3 given by Ben Niehoff. This will be extremely simple because the embedding is simple:

$$\phi(\theta,\phi)=\phi,\, \theta(\theta,\phi)=\theta,\,\psi(\theta,\phi)=const.$$

 

[mnb96]

arkajad!
Your explanation was extremely clear!
The procedure you apply seems to be analogous to the one used to find the line element in curvilinear coordinates. You basically start from the metric of our original space (Euclidean), then you define a parametric surface in it (the S^3 sphere) and you calculate the line element on it: this is supposed to be the pullback metric that the 3-sphere has "inherited" from the original space.

Funny that when I browsed some sources on the net related to pullback metric I found very little material, and most of it was inaccessible to me: I was led to think that the concept of pullback metric was something out of my range, simply too difficult to grasp for me. Now I suddenly realize that it is something that you could really teach and illustrate to a 6 years old child => Philosophical question: where was the fault? Is there perhaps a better way to teach maths/geometry? Or is it just me who needs to learn more abstract and sophisticated algebraic tools in order to understand such concepts in a more general framework?

I also read the articles you mentioned (through google translator). I was really surprised to see that this topic has beautiful connections with theoretical physics and twistor theory too. Believe it or not, my original post "hides" a problem related to computer-vision and the theory of color-spaces.

Thanks a lot!