English

On special types of minimal and totally geodesic unit vector fields

Differential Geometry 2007-05-23 v1

Abstract

We present a new equation with respect to a unit vector field on Riemannian manifold MnM^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 T1SnT_1S^n.

Keywords

Cite

@article{arxiv.math/0510002,
  title  = {On special types of minimal and totally geodesic unit vector fields},
  author = {Alexander Yampolsky},
  journal= {arXiv preprint arXiv:math/0510002},
  year   = {2007}
}

Comments

15 Pages

R2 v1 2026-07-22T17:25:16.647Z