  We consider a unit normal vector field of (local) hyperfoliation on a given
Riemannian manifold as a submanifold in the unit tangent bundle with Sasaki
metric. We give an explicit expression of the second fundamental form for this
submanifold and a rather simple condition its totally geodesic property in the
case of a totally umbilic hyperfoliation. A corresponding example shows the
non-triviality of this condition. In the 2-dimensional case, we give a complete
description of Riemannian manifolds admitting a geodesic unit vector field with
totally geodesic property.
