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 $K\ne 0,1$. 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 $-\frac{c^2}{4}$.
@article{119321,
author = {Yampolsky, Alexander L. Yampolsky, Alexander L.},
title = {On the intrinsic geometry of a unit vector field},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {43},
year = {2002},
pages = {299-317},
zbl = {1090.54013},
mrnumber = {1922129},
language = {en},
url = {http://dml.mathdoc.fr/item/119321}
}
Yampolsky, Alexander L., Yampolsky, Alexander L. On the intrinsic geometry of a unit vector field. Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002) pp. 299-317. http://gdmltest.u-ga.fr/item/119321/
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