The Kustaanheimo–Stiefel map, the Hopf fibration and the square root map on R3 and R4

We study the Kustaanheimo–Stiefel map (KSM) ψ from U∗ := R4 \ {0} to X∗ := R3 \ {0} and the principal circle bundle P = (U∗,ψ,X∗,S1) that it induces. We show that the KSM is the appropriate generalization of the squaring map z → z2, z ∈ C, and not quaternion-multiplication, in that the KSM induces a principal circle bundle on S3 → S2, namely the Hopf fibration, while quaternion-squaring is degenerate because the dimension of the fibers is not constant. We construct two square root branches from the upper and lower half of R3 to R3 \ (x1)− where (x1)− is the nonpositive x1-axis in R3 and resembles the cut used to define the standard complex square root branches ±√z. We glue these two branches together. We introduce what we like to call KS cylindrical coordinates with a 2-dimensional axis of rotation. We also introduce what we call KS torical and spherical coordinates. We use the KS cylindrical coordinates to define the full square root map on an S1-cover of R3 given by (R3 ×S1)/ ∼, where ∼ is an equivalence relation on (x1)− ×S1. © 2006 Elsevier Inc. All rights reserved.