Full description of totally geodesic unit vector fields on 2-dimensional Riemannian manifolds

TL;DR

This paper provides a comprehensive geometric characterization of totally geodesic unit vector fields on 2-dimensional Riemannian manifolds by examining their embeddings into the unit tangent bundle with the Sasaki metric.

Contribution

It offers a complete local description of such vector fields, linking their properties to the geometry of the manifold and its tangent bundle.

Findings

Characterization of totally geodesic unit vector fields

Embedding of 2D manifolds into their tangent bundles

Use of Sasaki metric for geometric analysis

Abstract

We give a full geometrical description of local totally geodesic unit vector field on Riemannian 2-manifold, considering the field as a local imbedding of the manifold into its unit tangent bundle with the Sasaki metric.