Geodesic
Full description of totally geodesic unit vector fields on 2-dimensional Riemannian manifolds
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 11 (2004) no. 3, pp. 355-365 Cet article a éte moissonné depuis la source Math-Net.Ru

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We give a full geometrical description of local totally geodesic unit vector field on Riemannian 2-manifold, considering the field as a local imbedding of the manifold into its unit tangent bundle with the Sasaki metric. 
@article{JMAG_2004_11_3_a7,
     author = {A. Yampolsky},
     title = {Full description of totally geodesic unit vector fields on 2-dimensional {Riemannian} manifolds},
     journal = {\v{Z}urnal matemati\v{c}eskoj fiziki, analiza, geometrii},
     pages = {355--365},
     year = {2004},
     volume = {11},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JMAG_2004_11_3_a7/}
}
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A. Yampolsky. Full description of totally geodesic unit vector fields on 2-dimensional Riemannian manifolds. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 11 (2004) no. 3, pp. 355-365. http://geodesic.mathdoc.fr/item/JMAG_2004_11_3_a7/