On the intrinsic geometry of a unit vector field

Yampolsky, Alexander L. Yampolsky, Alexander L.

Commentationes Mathematicae Universitatis Carolinae (2002)

  • Volume: 43, Issue: 2, page 299-317
  • ISSN: 0010-2628

Abstract

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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 ≠ 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 curvature - c 2 and of constant extrinsic curvature - c 2 4 .

How to cite

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Yampolsky, Alexander L., Yampolsky, Alexander L.. "On the intrinsic geometry of a unit vector field." Commentationes Mathematicae Universitatis Carolinae 43.2 (2002): 299-317. <http://eudml.org/doc/248993>.

@article{Yampolsky2002,
abstract = {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 $-\frac\{c^2\}\{4\}$.},
author = {Yampolsky, Alexander L., Yampolsky, Alexander L.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Sasaki metric; vector field; sectional curvature; totally geodesic submanifolds; unit vector field; Riemannian 2-manifold; tangent sphere bundle; Sasaki metric; sectional curvature; totally geodesic submanifolds},
language = {eng},
number = {2},
pages = {299-317},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On the intrinsic geometry of a unit vector field},
url = {http://eudml.org/doc/248993},
volume = {43},
year = {2002},
}

TY - JOUR
AU - Yampolsky, Alexander L., Yampolsky, Alexander L.
TI - On the intrinsic geometry of a unit vector field
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2002
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 43
IS - 2
SP - 299
EP - 317
AB - 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 $-\frac{c^2}{4}$.
LA - eng
KW - Sasaki metric; vector field; sectional curvature; totally geodesic submanifolds; unit vector field; Riemannian 2-manifold; tangent sphere bundle; Sasaki metric; sectional curvature; totally geodesic submanifolds
UR - http://eudml.org/doc/248993
ER -

References

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  2. Boeckx E., Vanhecke L., Harmonic and minimal radial vector fields, Acta Math. Hungar. 90 (2001), 317-331. (2001) Zbl1012.53040MR1910716
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  6. Gluck H., Ziller W., On the volume of a unit vector field on the three-sphere, Comment. Math. Helv. 61 (1986), 177-192. (1986) Zbl0605.53022MR0856085
  7. González-Dávila J.C., Vanhecke L., Examples of minimal unit vector fields, Ann. Global Anal. Geom. 18 (2000), 385-404. MR1795104
  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. (1971) Zbl0222.53044MR0286028
  9. Sasaki S., On the differential geometry of tangent bundles of Riemannian manifolds, Tôhoku Math. J. 10 (1958), 338-354. (1958) Zbl0086.15003MR0112152
  10. Klingenberg W, Sasaki S., Tangent sphere bundle of a 2 -sphere, Tôhoku Math. J. 27 (1975), 45-57. (1975) Zbl0309.53036MR0362149
  11. Yampolsky A., On the mean curvature of a unit vector field, Math. Publ. Debrecen, 2002, to appear. Zbl1010.53012MR1882460

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