On Special Types of Minimal and Totally Geodesic Unit Vector Fields

Abstract

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 Sasakian 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 $T_{1}S^{n}$.