On the intrinsic geometry of a unit vector field

2005/03/24 by Alexander Yampolsky
Mathematics · #math.DG #msc:53B25 #msc:53C25

paper · pdf

published as Comment. Mat. Univ. Carolinae 43, 2 (2002), 299-317 · 19 pages

arxiv created 2005/03/24 · arxiv updated 2009/12/01

Abstract

We study the geometrical properties of a unit vector field on a Riemannian 2-manifold, considering the field as a local imbedding of the manifold into its tangent sphere bundle with the Sasaki metric. For the case of constant curvature K, we give a description of the totally geodesic unit vector fields for K=0 and K=1 and prove a non-existence result for K not equal to 0 and 1. We also found a family of vector fields on the hyperbolic 2-plane L² of curvature -c² which generate foliations on unit tangent bundle over L² with leaves of constant intrinsic curvature -c² and of constant extrinsic curvature -c²/4.

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