The Hopf Fibration and its Applications

Among the most important topological constructions are Hopf fibrations. That they both play an important role in topology itself and have numerous physical applications lends credence to this statement. In this paper, I will discuss some of these applications as motivation for mathematicians to learn about the physical significance of the Hopf fibration in greater detail than that presented here. I will focus my attention primarily on rigid body rotation and the Hopf fibration h : S −→ S. Historically, the Hopf fibration was a cornerstone example that led to the development of modern differential geometry and gauge theory at the hands of people such as Chern. However, there is little mention in the mathematics literature of the fact that this example was already fundamentally present in rigid body mechanics! As I shall also explain, the Hopf fibration h is itself an example of what in mechanics is called a momentum map. To begin, I will briefly recall some definitions from algebra and topology. I will then define the Hopf fibration and look at three different cases (S −→ S, S −→ S, and S −→ S). After this overview, I shall narrow my attention to h : S −→ S and rigid body rotation. Briefly, the total spatial angular momentum of a rigid body is conserved in time. The set of positions and momenta of a rigid body with fixed spatial angular momentum is a copy of S/Z2. If one maps such points to their body angular momentum, the map h : S −→ S results. This map is exactly the Hopf fibration. Following a detailed discussion of this connection between the Hopf fibration and rigid body rotation, I will indicate the importance of the applications of this Hopf fibration. For several thousand years, mathematics and physics had coexisted in a symbiotic relationship that fostered numerous critical discoveries. Geometry was used to measure land in Egypt, while more sophisticated problems in surveying led Gauss to the intrinsic differential geometry of curves and surfaces in space. In particular, mathematics and classical mechanics have enjoyed a rather productive relationship. This symbiosis, however, weakened severely in the early 20th century before experiencing a strong revivial that has been especially prevalent during the past generation. This revival is largely due to Poincaré’s global geometric view. Topology and geometry have been shown to be intrinsic to the study of physics.