A double point calibration on an Sp(1) bundle over the three-sphere

SALVAI, MARCOS

JOURNAL OF DIFFERENTIAL GEOMETRY

INT PRESS BOSTON, INC

Año: 2001 vol. 59 p. 523 - 533

0022-040X

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 bun- dle 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 result- ing 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.