The affine approach to homogeneous geodesics in homogeneous Finsler spaces

Zdeněk Dušek

Archivum Mathematicum (2018)

  • Volume: 054, Issue: 5, page 257-263
  • ISSN: 0044-8753

Abstract

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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.

How to cite

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Dušek, Zdeněk. "The affine approach to homogeneous geodesics in homogeneous Finsler spaces." Archivum Mathematicum 054.5 (2018): 257-263. <http://eudml.org/doc/294673>.

@article{Dušek2018,
abstract = {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.},
author = {Dušek, Zdeněk},
journal = {Archivum Mathematicum},
keywords = {homogeneous space; Finsler space; Killing vector field; homogeneous geodesic},
language = {eng},
number = {5},
pages = {257-263},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {The affine approach to homogeneous geodesics in homogeneous Finsler spaces},
url = {http://eudml.org/doc/294673},
volume = {054},
year = {2018},
}

TY - JOUR
AU - Dušek, Zdeněk
TI - The affine approach to homogeneous geodesics in homogeneous Finsler spaces
JO - Archivum Mathematicum
PY - 2018
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 054
IS - 5
SP - 257
EP - 263
AB - 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.
LA - eng
KW - homogeneous space; Finsler space; Killing vector field; homogeneous geodesic
UR - http://eudml.org/doc/294673
ER -

References

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  2. Deng, S., Homogeneous Finsler Spaces, Springer Science+Business Media, New York, 2012. (2012) MR2962626
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  5. Dušek, Z., The existence of homogeneous geodesics in special homogeneous Finsler spaces, Matematički Vesnik (2018). (2018) MR3895904
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  7. Dušek, Z., Kowalski, O., Vlášek, Z., 10.1007/s00025-009-0373-1, Result. Math. 54 (2009), 273–288. (2009) MR2534447DOI10.1007/s00025-009-0373-1
  8. Figueroa-O’Farrill, J., Meessen, P., Philip, S., 10.1088/1126-6708/2005/05/050, J. High Energy Physics 05, 050 (2005). (2005) MR2155055DOI10.1088/1126-6708/2005/05/050
  9. Kowalski, O., Szenthe, J., 10.1023/A:1005287907806, Geom. Dedicata 81 (2000), 209–214, Erratum: Geom. Dedicata 84, 331–332 (2001). (2000) Zbl0980.53061MR1825363DOI10.1023/A:1005287907806
  10. Kowalski, O., Vanhecke, L., Riemannian manifolds with homogeneous geodesics, Boll. Un. Mat. Ital. B (7) 5 (1991), 189–246. (1991) Zbl0731.53046MR1110676
  11. Latifi, D., 10.1016/j.geomphys.2006.11.004, J. Geom. Phys. 57 (2007), 1421–1433. (2007) MR2289656DOI10.1016/j.geomphys.2006.11.004
  12. Yan, Z., 10.1007/s00605-016-0933-x, Monatsh. Math. 182 (1) (2017), 165–171. (2017) MR3592127DOI10.1007/s00605-016-0933-x
  13. Yan, Z., Huang, L., 10.1016/j.geomphys.2017.10.005, J. Geom. Phys. 124 (2018), 264–267. (2018) MR3754513DOI10.1016/j.geomphys.2017.10.005

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