| Title:
|
On the intrinsic geometry of a unit vector field (English) |
| Author:
|
Yampolsky, A. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
43 |
| Issue:
|
2 |
| Year:
|
2002 |
| Pages:
|
299-317 |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 $-\frac{c^2}{4}$. (English) |
| Keyword:
|
Sasaki metric |
| Keyword:
|
vector field |
| Keyword:
|
sectional curvature |
| Keyword:
|
totally geodesic submanifolds |
| MSC:
|
14E20 |
| MSC:
|
20C20 |
| MSC:
|
46E25 |
| MSC:
|
54C40 |
| idZBL:
|
Zbl 1090.54013 |
| idMR:
|
MR1922129 |
| . |
| Date available:
|
2009-01-08T19:22:10Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119321 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
[3] Boeckx E., Vanhecke L.: Harmonic and minimal vector fields on tangent and unit tangent bundles.Differential Geom. Appl. 13 (2000), 77-93. Zbl 0973.53053, MR 1775222 |
| Reference:
|
[4] Boeckx E., Vanhecke L.: Characteristic reflections on unit tangent sphere bundle.Houston J. Math. 23 (1997), 427-448. MR 1690045 |
| Reference:
|
[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 |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
[9] Sasaki S.: On the differential geometry of tangent bundles of Riemannian manifolds.Tôhoku Math. J. 10 (1958), 338-354. Zbl 0086.15003, MR 0112152 |
| Reference:
|
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| Reference:
|
[11] Yampolsky A.: On the mean curvature of a unit vector field.Math. Publ. Debrecen, 2002, to appear. Zbl 1010.53012, MR 1882460 |
| . |
