| 1. | Centre International de Rencontres Mathématiques-CIRM, 163 avenue de Luminy, Case 916, F-13288 Marseille -Cedex 9, France |
| 2. | Institut für Mathematik, Fakultät für Mathematik und Informatik, Friedrich-Schiller-Universität Jena, 07737 Jena, Germany |
We show how geodesics, Jacobi vector fields, and flag curvature of a Finsler metric behave under Zermelo deformation with respect to a Killing vector field. We also show that Zermelo deformation with respect to a Killing vector field of a locally symmetric Finsler metric is also locally symmetric.
| [1] |
D. Bao, C. Robles and Z. Shen,
Zermelo Navigation on Riemannian manifolds, J. Diff. Geom., 66 (2004), 377-435.
doi: 10.4310/jdg/1098137838. |
| [2] |
S. -S. Chern and Z. Shen, Riemann-Finsler Geometry, Nankai Tracts in Mathematics, 6, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. |
| [3] |
P. Foulon, Ziller-Katok deformations of Finsler metrics, in 2004 International Symposium on Finsler Geometry, Tianjin, PRC, 2004, 22-24. |
| [4] |
P. Foulon,
Locally symmetric Finsler spaces in negative curvature, C. R. Acad. Sci. Paris Sér. I Math., 324 (1997), 1127-1132.
doi: 10.1016/S0764-4442(97)87899-8. |
| [5] |
L. Huang and X. Mo,
On the flag curvature of a class of Finsler metrics produced by the navigation problem, Pac. J. Math., 277 (2015), 149-168.
doi: 10.2140/pjm.2015.277.149. |
| [6] |
M. A. Javaloyes and H. Vitório, Zermelo navigation in pseudo-Finsler metrics, preprint, arXiv: 1412.0465. |
| [7] |
A. B. Katok, Ergodic properties of degenerate integrable Hamiltonian systems, Izv. Akad. Nauk SSSR. Ser. Mat., 37 (1973), 539-576; English translation in Math. USSR-Isv. 7 (1973), 535-571. |
| [8] |
V. S. Matveev and M. Troyanov,
The Binet-Legendre metric in Finsler geometry, Geom. Topol., 16 (2012), 2135-2170.
doi: 10.2140/gt.2012.16.2135. |
| [9] |
Z. Shen, Finsler manifolds of constant positive curvature, in Finsler Geometry (Seattle, WA, 1995), Contemporary Math., 196, Amer. Math. Soc., Providence, RI, 1996, 83-93. |
| [10] |
Z. Shen, Differential Geometry of Spray and Finsler Spaces, Kluwer Academic Publishers, Dordrecht, 2001. |
show all references
| [1] |
D. Bao, C. Robles and Z. Shen,
Zermelo Navigation on Riemannian manifolds, J. Diff. Geom., 66 (2004), 377-435.
doi: 10.4310/jdg/1098137838. |
| [2] |
S. -S. Chern and Z. Shen, Riemann-Finsler Geometry, Nankai Tracts in Mathematics, 6, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. |
| [3] |
P. Foulon, Ziller-Katok deformations of Finsler metrics, in 2004 International Symposium on Finsler Geometry, Tianjin, PRC, 2004, 22-24. |
| [4] |
P. Foulon,
Locally symmetric Finsler spaces in negative curvature, C. R. Acad. Sci. Paris Sér. I Math., 324 (1997), 1127-1132.
doi: 10.1016/S0764-4442(97)87899-8. |
| [5] |
L. Huang and X. Mo,
On the flag curvature of a class of Finsler metrics produced by the navigation problem, Pac. J. Math., 277 (2015), 149-168.
doi: 10.2140/pjm.2015.277.149. |
| [6] |
M. A. Javaloyes and H. Vitório, Zermelo navigation in pseudo-Finsler metrics, preprint, arXiv: 1412.0465. |
| [7] |
A. B. Katok, Ergodic properties of degenerate integrable Hamiltonian systems, Izv. Akad. Nauk SSSR. Ser. Mat., 37 (1973), 539-576; English translation in Math. USSR-Isv. 7 (1973), 535-571. |
| [8] |
V. S. Matveev and M. Troyanov,
The Binet-Legendre metric in Finsler geometry, Geom. Topol., 16 (2012), 2135-2170.
doi: 10.2140/gt.2012.16.2135. |
| [9] |
Z. Shen, Finsler manifolds of constant positive curvature, in Finsler Geometry (Seattle, WA, 1995), Contemporary Math., 196, Amer. Math. Soc., Providence, RI, 1996, 83-93. |
| [10] |
Z. Shen, Differential Geometry of Spray and Finsler Spaces, Kluwer Academic Publishers, Dordrecht, 2001. |
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