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

TL;DR

This paper explores the decomposition of rotations in 3D and 4D spaces to analyze stereographic projections of the Clifford torus, revealing a unique symmetry property that helps characterize it among minimal tori in the 3-sphere.

Contribution

It introduces a novel decomposition approach for rotations in R^3 and R^4 that elucidates the symmetry of stereographic projections of the Clifford torus, a first in the field.

Findings

Identifies a new symmetry property of stereographic projections of the Clifford torus.

Provides a decomposition formula for rotations with respect to stereographic projection.

Suggests a potential characterization of the Clifford torus among minimal tori in S^3.

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.