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.
@article{1113247180,
author = {Gil-Medrano, Olga and Llinares-Fuster, Elisa},
title = {Minimal unit vector fields},
journal = {Tohoku Math. J. (2)},
volume = {54},
number = {1},
year = {2002},
pages = { 71-84},
language = {en},
url = {http://dml.mathdoc.fr/item/1113247180}
}
Gil-Medrano, Olga; Llinares-Fuster, Elisa. Minimal unit vector fields. Tohoku Math. J. (2), Tome 54 (2002) no. 1, pp. 71-84. http://gdmltest.u-ga.fr/item/1113247180/