LEFT-INVARIANT MINIMAL UNIT VECTOR FIELDS ON A LIE GROUP OF CONSTANT NEGATIVE SECTIONAL CURVATURE

Abstract

Abstract. We find all left-invariant minimal unit vector fields and stro-ngly normal unit vector fields on a Lie group which is isometric to thehyperbolic space. 1. IntroductionA smooth unit vector field on a Riemannian manifold ( M,g ) is a cross sectionof its unit sphere bundle T 1 ( M ) and hence can be viewed as a submanifold of T 1 ( M ). If the manifold M is compact and T 1 ( M ) is equipped with a naturalRiemannian metric g s called the Sasaki metric, then the volume of the unitvector field is defined as the volume of this submanifold.For the problem of determining unit vector fields which have minimal vol-ume, Gluck and Ziller showed that the unit vector fields of minimal volume on S 3 are precisely the Hopf vector fields and no others ([7]). But in the higherdimensional spheres, S 2 n +1 ,k ≥ 2, this is not the case ([4], [8], [10]).The problem of finding unit vector fields of the minimum volume seems tobe very difficult, so it is natural to consider the problem of finding the criticalvalues or critical points of the volume functional.Gil-Medriano and Llinares-Fuster proved that a unit vector field is a criticalpoint of the volume functional if and only if the corresponding immersion in(