The Hopf Fibration
As a nontrivial example of a foliation, I present the “Hopf fibration”. The name I won’t really explain quite yet, but we’ll see it’s a one-dimensional foliation of the three-dimensional sphere.
So, first let’s get our hands on the three-sphere . This is by definition the collection of vectors of length in , but I want to consider this definition slightly differently. Since we know that the complex plane is isomorphic to the real plane as a real vector space, so we find the isomorphism . Now we use the inner product on to define as the collection of vectors with .
Now for each we can define a foliation. The leaf through the point is the curve . Since multiplying by and doesn’t change the norm of a complex number, this whole curve is still contained within . Every point in is clearly contained in some such curve, and two points being contained within the same curve is an equivalence relation: any point is in the same curve as itself; if and , then and ; and if , , and , then and . This shows that the curves do indeed partition .
Now we need to show that the tangent spaces to the leaves provide a distribution on . Since this will be a one-dimensional distribution, we just need to find an everywhere nonzero vector field tangent to the leaves, and the derivative of the curve through each point will do nicely. At we get the derivative
It should be clear that this defines a smooth vector field over all of , though it may not be clear from the formulas that these vectors are actually tangent to . To see this we can either (messily) convert back to real coordinates or we can think geometrically and see that the tangent to a curve within a submanifold must be tangent to that submanifold.
The Hopf fibration is what results when we pick , but the case of irrational is very interesting. In this case we find that some leaves curve around and meet themselves, forming circles, while others never meet themselves, forming homeomorphic images of the whole real line. What this tells us is that not all the leaves of a foliation have to look like each other.
To see this, we try to solve the equations
The first equation tells us that either or . In the first case, we simply have the circle . In the second case, the second equation tells us that either or . The case where is similar to the case , but if neither coordinate is zero then we find . But we assumed that is irrational, so we get no nontrivial solutions for here.
Since the curves don’t change the length of either component, we can get other examples of foliations. For instance, if we let , then the curve will stay on the torus where each circle has radius in its copy of . Looking at all the curves on this surface gives a foliation of the torus. If is irrational, the curve winds around and around the donut-shaped surface, never quite coming back to touch itself, but eventually coming arbitrarily close to any given point on the surface.
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