Previous |  Up |  Next


Sasaki metric; vector field; sectional curvature; totally geodesic submanifolds
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 $-\frac{c^2}{4}$.
[1] Aminov Yu.: The Geometry of Vector Fields. Gordon & Breach Publ., 2000. MR 1749926 | Zbl 0965.53002
[2] Boeckx E., Vanhecke L.: Harmonic and minimal radial vector fields. Acta Math. Hungar. 90 (2001), 317-331. MR 1910716 | Zbl 1012.53040
[3] Boeckx E., Vanhecke L.: Harmonic and minimal vector fields on tangent and unit tangent bundles. Differential Geom. Appl. 13 (2000), 77-93. MR 1775222 | Zbl 0973.53053
[4] Boeckx E., Vanhecke L.: Characteristic reflections on unit tangent sphere bundle. Houston J. Math. 23 (1997), 427-448. MR 1690045
[5] Borisenko A., Yampolsky A.: The sectional curvature of the Sasaki metric of $T_1M^n$. Ukrain. Geom. Sb. 30 (1987), 10-17 English transl.: J. Soviet Math. 51 (1990), 5 2503-2508. MR 0914771
[6] Gluck H., Ziller W.: On the volume of a unit vector field on the three-sphere. Comment. Math. Helv. 61 (1986), 177-192. MR 0856085 | Zbl 0605.53022
[7] González-Dávila J.C., Vanhecke L.: Examples of minimal unit vector fields. Ann. Global Anal. Geom. 18 (2000), 385-404. MR 1795104
[8] Kowalski O.: Curvature of the induced Riemannian metric on the tangent bundle of a Riemannian manifold. J. Reine Angew. Math. 250 (1971), 124-129. MR 0286028 | Zbl 0222.53044
[9] Sasaki S.: On the differential geometry of tangent bundles of Riemannian manifolds. Tôhoku Math. J. 10 (1958), 338-354. MR 0112152 | Zbl 0086.15003
[10] Klingenberg W, Sasaki S.: Tangent sphere bundle of a $2$-sphere. Tôhoku Math. J. 27 (1975), 45-57. MR 0362149 | Zbl 0309.53036
[11] Yampolsky A.: On the mean curvature of a unit vector field. Math. Publ. Debrecen, 2002, to appear. MR 1882460 | Zbl 1010.53012
Partner of
EuDML logo