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

We study the Kustaanheimo-Stiefel map (KSM) psi from U* := R4 \ {0} to X* := R3 \ {0} and the principal circle bundle P = (U*, psi, X*, S-1) that it induces. We show that the KSM is the appropriate generalization of the squaring map z -> z(2), z is an element of C, and not quaternion-multiplication, in that the KSM induces a principal circle bundle on S-3 -> S-2, namely the Hopf fibration, while quatermon-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 R-3 to R-3 (x(1))(-) where (x(1))(-) is the nonpositive x(1)-axis in R-3 and resembles the cut used to define the standard complex square root branches +/-root 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 S-1-cover of R-3 given by (R-3 x S-1)/similar to, where similar to is an equivalence relation on (x(1))(-) x S-1. (c) 2006 Elsevier Inc. All rights reserved.