A Note on Skew-Hopf Fibrations

Abstract

Each great circle fibration of the unit $3$-sphere in $4$-space can be identified with a subset of the Grassmann manifold of oriented $2$-planes in $4$-space by associating each great circle fiber with the $2$-plane it lies in. This Grassmann manifold can be identified with the space ${S^2} \times {S^2}$. H. Gluck and F. Warner, in Great circle fibrations of the three sphere, Duke Math. J. 50 (1983), 107-132, have shown that the subsets of this Grassmann manifold which correspond to great circle fibrations can be interpreted as the graphs of distance decreasing maps from ${S^2}$ and ${S^2}$ and that Hopf fibrations correspond to constant maps. This note characterizes explicitly the maps which correspond to "skew-Hopf" fibrations: those fibrations of the $3$-sphere obtained from Hopf fibrations by applying a linear transformation of $4$-space followed by projection of the fibers back to the unit $3$-sphere.