| Title:
|
The affine approach to homogeneous geodesics in homogeneous Finsler spaces (English) |
| Author:
|
Dušek, Zdeněk |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
54 |
| Issue:
|
5 |
| Year:
|
2018 |
| Pages:
|
257-263 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In the recent paper [Yan, Z.: Existence of homogeneous geodesics on homogeneous Finsler spaces of odd dimension, Monatsh. Math. 182,1, 165–171 (2017)], it was claimed that any homogeneous Finsler space of odd dimension admits a homogeneous geodesic through any point. However, the proof contains a serious gap. The situation is a bit delicate, because the statement is correct. In the present paper, the incorrect part in this proof is indicated. Further, it is shown that homogeneous geodesics in homogeneous Finsler spaces can be studied by another method developed in earlier works by the author for homogeneous affine manifolds. This method is adapted for Finsler geometry and the statement is proved correctly. (English) |
| Keyword:
|
homogeneous space |
| Keyword:
|
Finsler space |
| Keyword:
|
Killing vector field |
| Keyword:
|
homogeneous geodesic |
| MSC:
|
53C22 |
| MSC:
|
53C30 |
| MSC:
|
53C60 |
| idZBL:
|
Zbl 06997354 |
| idMR:
|
MR3887353 |
| DOI:
|
10.5817/AM2018-5-257 |
| . |
| Date available:
|
2018-12-06T16:12:43Z |
| Last updated:
|
2020-01-05 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147503 |
| . |
| Reference:
|
[1] Bao, D., Chern, S.-S., Shen, Z.: An Introduction to Riemann-Finsler Geometry.Springer Science+Business Media, New York, 2000. Zbl 0954.53001, MR 1747675 |
| Reference:
|
[2] Deng, S.: Homogeneous Finsler Spaces.Springer Science+Business Media, New York, 2012. MR 2962626 |
| Reference:
|
[3] Dušek, Z.: On the reparametrization of affine homogeneous geodesics.Differential Geometry, Proceedings of the VIII International Colloquium World Scientific (Singapore) (López, J.A. Álvarez, García-Río, E., eds.), 2009, pp. 217–226. MR 2523507 |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
|
[12] Yan, Z.: Existence of homogeneous geodesics on homogeneous Finsler spaces of odd dimension.Monatsh. Math. 182 (1) (2017), 165–171. MR 3592127, 10.1007/s00605-016-0933-x |
| Reference:
|
[13] Yan, Z., Huang, L.: On the existence of homogeneous geodesic in homogeneous Finsler space.J. Geom. Phys. 124 (2018), 264–267. MR 3754513, 10.1016/j.geomphys.2017.10.005 |
| . |
