Gluck and Ziller proved that Hopf vector fields on S3 have minimum volume among all unit vector fields. Thinking of S3 as a Lie group, Hopf vector fields are exactly those with unit length which are left or right invariant, and TS3 is a trivial vector bundle with a connection induced by the adjoint representation. We prove the analogue of the stated result of Gluck and Ziller for the representation given by quaternionic multiplication. The resulting vector bundle over S3, with the Sasaki metric, has as well no parallel unit sections. We provide an application of a double point calibration, proving that the submanifolds determined by the left and right invariant sections minimize volume in their homology classes.
"A Two Point Calibration on an SP(1) Bundle Over the Three-Sphere." J. Differential Geom. 59 (3) 523 - 533, November, 2001. https://doi.org/10.4310/jdg/1090349450