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. We nd all left-invariant minimal unit vector elds and stro-ngly normal unit vector elds on a Lie group which is isometric to thehyperbolic space. 1. IntroductionA smooth unit vector eld 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 eld is dened as the volume of this submanifold.For the problem of determining unit vector elds which have minimal vol-ume, Gluck and Ziller showed that the unit vector elds of minimal volume on S 3 are precisely the Hopf vector elds 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 nding unit vector elds of the minimum volume seems tobe very dicult, so it is natural to consider the problem of nding the criticalvalues or critical points of the volume functional.Gil-Medriano and Llinares-Fuster proved that a unit vector eld is a criticalpoint of the volume functional if and only if the corresponding immersion in(

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