  We give a complete list of those left invariant unit vector fields on
three-dimensional Lie groups with the left-invariant metric that generate a
totally geodesic submanifold in the unit tangent bundle of a group with the
Sasaki metric. As a result, each class of three-dimensional Lie groups admits
the totally geodesic unit vector field. From geometrical viewpoint, the field
is either parallel or characteristic vector field of a natural almost contact
structure on the group.
