# 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

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topYampolsky, 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 -

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