
-
Previous Article
Hyperbolic dynamics of discrete dynamical systems on pseudo-riemannian manifolds
- ERA-MS Home
- This Volume
- Next Article
Zermelo deformation of finsler metrics by killing vector fields
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.
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. 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. |
[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. 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. |
[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
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. 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. |
[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. 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. |
[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. |

[1] |
Jean-François Biasse. Improvements in the computation of ideal class groups of imaginary quadratic number fields. Advances in Mathematics of Communications, 2010, 4 (2) : 141-154. doi: 10.3934/amc.2010.4.141 |
[2] |
Mao Okada. Local rigidity of certain actions of solvable groups on the boundaries of rank-one symmetric spaces. Journal of Modern Dynamics, 2021, 17: 111-143. doi: 10.3934/jmd.2021004 |
[3] |
Yves Dumont, Frederic Chiroleu. Vector control for the Chikungunya disease. Mathematical Biosciences & Engineering, 2010, 7 (2) : 313-345. doi: 10.3934/mbe.2010.7.313 |
[4] |
Alexandr Mikhaylov, Victor Mikhaylov. Dynamic inverse problem for Jacobi matrices. Inverse Problems & Imaging, 2019, 13 (3) : 431-447. doi: 10.3934/ipi.2019021 |
[5] |
A. K. Misra, Anupama Sharma, Jia Li. A mathematical model for control of vector borne diseases through media campaigns. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1909-1927. doi: 10.3934/dcdsb.2013.18.1909 |
[6] |
F.J. Herranz, J. de Lucas, C. Sardón. Jacobi--Lie systems: Fundamentals and low-dimensional classification. Conference Publications, 2015, 2015 (special) : 605-614. doi: 10.3934/proc.2015.0605 |
[7] |
Feng Luo. A combinatorial curvature flow for compact 3-manifolds with boundary. Electronic Research Announcements, 2005, 11: 12-20. |
[8] |
Alexandre B. Simas, Fábio J. Valentim. $W$-Sobolev spaces: Higher order and regularity. Communications on Pure & Applied Analysis, 2015, 14 (2) : 597-607. doi: 10.3934/cpaa.2015.14.597 |
[9] |
Alexander A. Davydov, Massimo Giulietti, Stefano Marcugini, Fernanda Pambianco. Linear nonbinary covering codes and saturating sets in projective spaces. Advances in Mathematics of Communications, 2011, 5 (1) : 119-147. doi: 10.3934/amc.2011.5.119 |
[10] |
Xue-Ping Luo, Yi-Bin Xiao, Wei Li. Strict feasibility of variational inclusion problems in reflexive Banach spaces. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2495-2502. doi: 10.3934/jimo.2019065 |
[11] |
Guido De Philippis, Antonio De Rosa, Jonas Hirsch. The area blow up set for bounded mean curvature submanifolds with respect to elliptic surface energy functionals. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 7031-7056. doi: 10.3934/dcds.2019243 |
[12] |
Charles Fulton, David Pearson, Steven Pruess. Characterization of the spectral density function for a one-sided tridiagonal Jacobi matrix operator. Conference Publications, 2013, 2013 (special) : 247-257. doi: 10.3934/proc.2013.2013.247 |
[13] |
Lei Liu, Li Wu. Multiplicity of closed characteristics on $ P $-symmetric compact convex hypersurfaces in $ \mathbb{R}^{2n} $. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020378 |
[14] |
Ademir Fernando Pazoto, Lionel Rosier. Uniform stabilization in weighted Sobolev spaces for the KdV equation posed on the half-line. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1511-1535. doi: 10.3934/dcdsb.2010.14.1511 |
[15] |
Lucas C. F. Ferreira, Jhean E. Pérez-López, Élder J. Villamizar-Roa. On the product in Besov-Lorentz-Morrey spaces and existence of solutions for the stationary Boussinesq equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2423-2439. doi: 10.3934/cpaa.2018115 |
[16] |
Ka Luen Cheung, Man Chun Leung. Asymptotic behavior of positive solutions of the equation $ \Delta u + K u^{\frac{n+2}{n-2}} = 0$ in $IR^n$ and positive scalar curvature. Conference Publications, 2001, 2001 (Special) : 109-120. doi: 10.3934/proc.2001.2001.109 |
2019 Impact Factor: 0.5
Tools
Metrics
Other articles
by authors
[Back to Top]