On special types of minimal and totally geodesic unit vector fields

TL;DR

This paper introduces a new equation characterizing unit vector fields that generate totally geodesic submanifolds in the tangent bundle, identifies the Hopf vector field as unique in a specific class, and provides conditions for geodesic fields on spheres.

Contribution

It presents a novel equation for totally geodesic unit vector fields, characterizes the Hopf vector field's uniqueness, and establishes new criteria for geodesic fields on spheres.

Findings

Hopf vector field is unique among covariantly normal unit vector fields with totally geodesic property.

New necessary and sufficient condition for geodesic unit vector fields on spheres to generate totally geodesic submanifolds.

Introduces a new equation linking unit vector fields to totally geodesic submanifolds in the tangent bundle.

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 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 $T_1S^n$.