On the volume of a unit vector field on the three-sphere. (English) Zbl 0605.53022

Gluck, Herman; Ziller, Wolfgang

Comment. Math. Helv. 61, 177-192 (1986).

A unit vector field on a compact Riemannian manifold can be considered to be a cross-section, and hence submanifold, of the unit tangent bundle. The volume of a vector field is defined to be the volume of this submanifold, measured in the natural Riemannian metric on the unit tangent bundle. One hopes that the ”visually best organized” vector fields have the minimum possible volume.
Using the method of calibrated geometries one proves that the unit vector fields of minimum volume on \(S^ 3\) are the Hopf vector fields, and no others, where a Hopf vector field is the unit vector field tangent to the Hopf fibration of \(S^ 3\) over \(S^ 2\), and all those vector fields congruent to this one. This method fails in higher dimensions leaving the question as to whether the Hopf vector fields on higher odd dimensional spheres minimize volume or not.

Reviewer: J.Hebda

MSC:

53C20	Global Riemannian geometry, including pinching
53C65	Integral geometry
53C40	Global submanifolds

Keywords:

vector field; calibrated geometries; minimum volume; Hopf fibration