Section 6 Hopf Fibration/Clifford Parallels

Figure 1: Cutaway perspective view in R³ of the Hopf fibration of S₃ over S₂: Visualization via stereographic projection of the foliation of S₃ —>R⁴ by nested flat tori. The Hopf fibration together with the bundle homomorphism S₃ —> S0(3, R) forms a fiber bundle model of the symplectic spin structure on the bitangent bundle of S₂, The traces of the bitangential planes on the flat tori are displayed as a bundle of Villarceau circles of parataxy. The Hopf bundle is tautologous in the sense that a point in P₁(C ) represents a complex line in the Hermitian vector space C +C (tensor addition), and to this point its complex line is associated. By fiberwise multiplication with the time-line R, the four-dimensional Taub-NUT manifold arises which forms a cylinder bundle over S₂-

light echo phenomenon of extragalactic quantum holography with radiation from the core of the galaxis serving as the reference radiation.