  We present a new equation with respect to a unit vector field on Riemannian
manifold $M^n$ such that its solution defines a totally geodesic submanifold in
the unit tangent bundle with Sasaki metric and apply it to some classes of unit
vector fields. We introduce a class of covariantly normal unit vector fields
and prove that within this class the Hopf vector field is a unique global one
with totally geodesic property. For the wider class of geodesic unit vector
fields on a sphere we give a new necessary and sufficient condition to generate
a totally geodesic submanifold in $T_1S^n$.
