  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 L^2 of curvature
-c^2 which generate foliations on unit tangent bundle over L^2 with leaves of
constant intrinsic curvature -c^2 and of constant extrinsic curvature -c^2/4.
