Rotations of the three-sphere and symmetry of the Clifford torus

Abstract

We describe decomposition formulas for rotations of $R^3$ and $R^4$ that have special properties with respect to stereographic projection. We use the lower dimensional decomposition to analyze stereographic projections of great circles in $S^2 \subset R^3$. This analysis provides a pattern for our analysis of stereographic projections of the Clifford torus ${\mathcal C}\subset S^3 \subset R^4$. We use the higher dimensional decomposition to prove a symmetry assertion for stereographic projections of ${\mathcal C}$ which we believe we are the first to observe and which can be used to characterize the Clifford torus among embedded minimal tori in $S^3$---though this last assertion goes beyond the scope of this paper. An effort is made to intuitively motivate all necessary concepts including rotation, stereographic projection, and symmetry.