On special types of minimal and totally geodesic unit vector fields

2005/09/30 by Alexander Yampolsky, Yampolsky, Alexander
Mathematics · Physics and Astronomy · #53B20 #53B25 #53C25 #Advanced Differential Equations and Dynamical Systems #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.48550/arxiv.math/0510002

openalex publication_date 2005/09/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We present a new equation with respect to a unit vector field on Riemannian manifold Mⁿ 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 T₁Sⁿ.

Related

• Totally geodesic property of the Hopf vector field
• Full description of totally geodesic unit vector fields on 2-dimensional Riemannian manifolds
• Invariant totally geodesic unit vector fields on three-dimensional Lie groups
• On extrinsic geometry of unit normal vector fields of Riemannian hyperfoliations
• Transverse totally geodesic submanifolds of the tangent bundle