M ar 2 00 5 On the intrinsic geometry of a unit vector field ∗

We study the geometrical properties of a unit vector field on a Riemann-ian 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 = 0, 1. We also found a family ξω of vector fields on the hyperbolic 2-plane L 2 of curvature −c 2 which generate foliations on T1L 2 with leaves of constant intrinsic curvature −c 2 and of constant extrinsic curvature − c 2 4 .