More Hopf Fibrations

  1. viz 1 (available in VR)
  2. viz 2 (available in VR)

The Hopf Flow

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For an introduction to the Hopf Fibration, see our first post on the Hopf Fibration.

If we look at different algebraic structures on the three-sphere, we discover new aspects of the Hopf fibration.

For instance, we can view four-dimensional space as a complex vector space in two complex dimensions. Then, the three-sphere can be recognized as all of the points \((z0,z1)\) with \(|z0|^2 + |z1|^2 = 1\). The fibers of the Hopf fibration are then the intersection of all the complex lines with the three-sphere. That is to say, a fiber of the Hopf fibration can be defined as all points on the three-sphere satisfying \(z0/z1 = c\), for some constant \(c\).

With this setup in mind, we see that the scalar multiplication by \(e^{it}\) preserves fibers, and is in fact an isometry of the three-sphere -- it preserves distances between points. This is a one-parameter family of isometries connecting the identity map with scalar multiplication by \(i\). It is known as the Hopf flow and is visualized in the demonstration above when setting type = "left".

Each Hopf fiber corresponds to a great circle, a circle of maximal size on the three-sphere. In the visualization, each particle flows along its Hopf fiber. Particles of the same color belong to the same Hopf fiber. Every point flows along its own great circle and moves in a similar way to every other point. This distinguishes the three-sphere from the two-sphere as all of the flows of the two-sphere are rotational, and therefore has fixed points.

If we place image of the fibers to be a circle near the north pole, we see what it feel like to float in the three-sphere and be pushed along the Hopf flow. Our bodies would rotate clockwise when facing the direction of the Hopf flow as we orbited around the three-sphere along a great circle.

Looking at another algebraic structure yields yet more interesting phenomena. We can view four-dimensional space as the space of quaternions. The unit quaternions form a group, and left multiplication by the quaternion \(i\) corresponds to scalar multiplication by the complex number \(i\). Right multiplication by a unit quaternion also preserves the group, and is an isometry of the three-sphere. Right multiplication by the quaternion \(i\) has the same effect as left multiplication by the complex scalar \(i\) on the first complex coordinate of \(\C^2\), but has the effect of multiplying by the complex scalar \(-i\) on the second complex coordinate. We see that the one-parameter family of isometries for right multiplication by \(i\) corresponds to multiplying the first complex coordinate by \(e^{it}\) and the second complex coordinate by \(e^{-it}\). This is called the right Hopf flow and can be visualized above by selecting type = "right".

In general, the right Hopf flow no longer preserves fibers. Instead fibers are mapped to neighboring fibers corresponding to the same latitude on the two-sphere.

The right and left Hopf flows are in general not parallel. The only exception is at the fibers corresponding to the north and south poles of the two-sphere. At the south pole, the Hopf flows are equal, and at the north pole, they are opposites.

Orthogonal Hopf Fibrations

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We can use any pure unit quaternion for the role of the complex scalar \(i\). In particular, we can replace the quaternion \(i\) with \(j\) or \(k\), and obtain two other Hopf fibrations. Furthermore, the three Hopf fibrations corresponding to i, j and k are mutually orthogonal. These orthogonal Hopf fibrations are visualized in the second math demonstration above. Having all three fibrations displayed at once can clutter the display, so clicking on the display button will cycle through the three Hopf fibrations.

To reveal the symmetry between the three Hopf fibrations, the stereographic projection used here projects from the point \((-1,0,0,0)\), which sends 1 to the origin, and \(i,j,k\) to the coordinate axes.


Material for this math demonstration comes from Chapter 2.7 of Bill Thurston's Three-Dimensional Geometry and Topology.