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 $K\not =0,1$. We also found a family $\xi _\omega $ of vector fields on the hyperbolic 2-plane $L^2$ of curvature $-c^2$ which generate foliations on $T_1L^2$ with leaves of constant intrinsic curvature $-c^2$ and of constant extrinsic curvature $-\frac {c^2}{4}$.
Keywords: Sasaki metric, vector field, sectional curvature, totally geodesic submanifolds
AMS Subject Classification: Primary 54C40, 14E20; Secondary 46E25, 20C20