Tohoku Mathematical Journal
2002

March
SECOND SERIES VOL. 54, NO. 1

Tohoku Math. J.
54 (2002), 71-84

Title MINIMAL UNIT VECTOR FIELDS

Author Olga Gil-Medrano and Elisa Llinares-Fuster

(Received March 27, 2000, revised April 4, 2001)
Abstract. We compute the first variation of the functional that assigns each unit vector field the volume of its image in the unit tangent bundle. It is shown that critical points are exactly those vector fields that determine a minimal immersion. We also find a necessary and sufficient condition that a vector field, defined in an open manifold, must fulfill to be minimal, and obtain a simpler equivalent condition when the vector field is Killing. The condition is fulfilled, in particular, by the characteristic vector field of a Sasakian manifold and by Hopf vector fields on spheres.

2000 Mathematics Subject Classification. Primary 53C20; Secondary 53C25, 53C42.

Key words and phrases. Volume of vector fields, critical points, minimal vector fields, Killing vector fields, Hopf vector fields, Sasakian manifolds.

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