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On the intrinsic geometry of a unit vector field

Yampolsky, Alexander L. Yampolsky, Alexander L. — 2002

Commentationes Mathematicae Universitatis Carolinae

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 $K \neq 0, 1$ . We also found a family $ξ_{ω}$ of vector fields on the hyperbolic 2-plane $L^{2}$ of curvature $- c^{2}$ which generate foliations on $T_{1} L^{2}$ with leaves of constant intrinsic...

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