Geodesic
On the intrinsic geometry of a unit vector field
Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002) no. 2, pp. 299-317.

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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 $-\frac{c^2}{4}$.
Classification : 14E20, 20C20, 46E25, 54C40
Keywords: Sasaki metric; vector field; sectional curvature; totally geodesic submanifolds 
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     title = {On the intrinsic geometry of a unit vector field},
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Yampolsky, A. On the intrinsic geometry of a unit vector field. Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002) no. 2, pp. 299-317. http://geodesic.mathdoc.fr/item/CMUC_2002__43_2_a8/