Zermelo deformation of finsler metrics by killing vector fields

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March 2018, 25: 1-7. doi: 10.3934/era.2018.25.001

Zermelo deformation of finsler metrics by killing vector fields

Patrick Foulon ^1, and Vladimir S. Matveev ^2,

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

Received October 10, 2017 Published March 2018

Fund Project: The authors thank Sergei Ivanov for useful comments. V. M. was partially supported by the University of Jena and by the DFG grant MA 2565/4

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.

Keywords: Finsler, Zermelo deformation, Killing vector fields, Jacobi fields, curvature, symmetric spaces.

Mathematics Subject Classification: Primary: 53B40. Secondary: 53C35.

Citation: Patrick Foulon, Vladimir S. Matveev. Zermelo deformation of finsler metrics by killing vector fields. Electronic Research Announcements, 2018, 25: 1-7. doi: 10.3934/era.2018.25.001

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. Google Scholar
[2]	S. -S. Chern and Z. Shen, Riemann-Finsler Geometry, Nankai Tracts in Mathematics, 6, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. Google Scholar
[3]	P. Foulon, Ziller-Katok deformations of Finsler metrics, in 2004 International Symposium on Finsler Geometry, Tianjin, PRC, 2004, 22-24.Google Scholar
[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. Google Scholar
[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. Google Scholar
[6]	M. A. Javaloyes and H. Vitório, Zermelo navigation in pseudo-Finsler metrics, preprint, arXiv: 1412.0465.Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[10]	Z. Shen, Differential Geometry of Spray and Finsler Spaces, Kluwer Academic Publishers, Dordrecht, 2001. Google Scholar

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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. Google Scholar
[2]	S. -S. Chern and Z. Shen, Riemann-Finsler Geometry, Nankai Tracts in Mathematics, 6, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. Google Scholar
[3]	P. Foulon, Ziller-Katok deformations of Finsler metrics, in 2004 International Symposium on Finsler Geometry, Tianjin, PRC, 2004, 22-24.Google Scholar
[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. Google Scholar
[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. Google Scholar
[6]	M. A. Javaloyes and H. Vitório, Zermelo navigation in pseudo-Finsler metrics, preprint, arXiv: 1412.0465.Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[10]	Z. Shen, Differential Geometry of Spray and Finsler Spaces, Kluwer Academic Publishers, Dordrecht, 2001. Google Scholar

Figure 1. The unit ball of $\tilde F$ (dashed line) is the $v$-translation of that of $F$ (bold line). If a vector $J$ is tangent to the unit ball of $ F$ at $\xi$, it is tangent to the unit ball of $ \tilde F$ at $\xi + v$

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