American Institute of Mathematical Sciences

Advanced
  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

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.  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 $\xi + v$
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Ioan Bucataru. A setting for higher order differential equation fields and higher order Lagrange and Finsler spaces. Journal of Geometric Mechanics, 2013, 5 (3) : 257-279. doi: 10.3934/jgm.2013.5.257
[2]

Siamak RabieniaHaratbar. Support theorem for the Light-Ray transform of vector fields on Minkowski spaces. Inverse Problems & Imaging, 2018, 12 (2) : 293-314. doi: 10.3934/ipi.2018013
[3]

Leonardo Câmara, Bruno Scárdua. On the integrability of holomorphic vector fields. Discrete & Continuous Dynamical Systems, 2009, 25 (2) : 481-493. doi: 10.3934/dcds.2009.25.481
[4]

Jifeng Chu, Zhaosheng Feng, Ming Li. Periodic shadowing of vector fields. Discrete & Continuous Dynamical Systems, 2016, 36 (7) : 3623-3638. doi: 10.3934/dcds.2016.36.3623
[5]

Hitoshi Ishii, Taiga Kumagai. Averaging of Hamilton-Jacobi equations along divergence-free vector fields. Discrete & Continuous Dynamical Systems, 2021, 41 (4) : 1519-1542. doi: 10.3934/dcds.2020329
[6]

BronisŁaw Jakubczyk, Wojciech Kryński. Vector fields with distributions and invariants of ODEs. Journal of Geometric Mechanics, 2013, 5 (1) : 85-129. doi: 10.3934/jgm.2013.5.85
[7]

Davi Obata. Symmetries of vector fields: The diffeomorphism centralizer. Discrete & Continuous Dynamical Systems, 2021, 41 (10) : 4943-4957. doi: 10.3934/dcds.2021063
[8]

Livio Flaminio, Miguel Paternain. Linearization of cohomology-free vector fields. Discrete & Continuous Dynamical Systems, 2011, 29 (3) : 1031-1039. doi: 10.3934/dcds.2011.29.1031
[9]

Alexander Krasnosel'skii, Jean Mawhin. The index at infinity for some vector fields with oscillating nonlinearities. Discrete & Continuous Dynamical Systems, 2000, 6 (1) : 165-174. doi: 10.3934/dcds.2000.6.165
[10]

José Luis Bravo, Manuel Fernández, Ignacio Ojeda, Fernando Sánchez. Uniqueness of limit cycles for quadratic vector fields. Discrete & Continuous Dynamical Systems, 2019, 39 (1) : 483-502. doi: 10.3934/dcds.2019020
[11]

Antoni Ferragut, Jaume Llibre, Adam Mahdi. Polynomial inverse integrating factors for polynomial vector fields. Discrete & Continuous Dynamical Systems, 2007, 17 (2) : 387-395. doi: 10.3934/dcds.2007.17.387
[12]

Jorge Sotomayor, Michail Zhitomirskii. On pairs of foliations defined by vector fields in the plane. Discrete & Continuous Dynamical Systems, 2000, 6 (3) : 741-749. doi: 10.3934/dcds.2000.6.741
[13]

Benedetto Piccoli, Francesco Rossi. Measure dynamics with Probability Vector Fields and sources. Discrete & Continuous Dynamical Systems, 2019, 39 (11) : 6207-6230. doi: 10.3934/dcds.2019270
[14]

Jaume Llibre, Claudia Valls. Centers for polynomial vector fields of arbitrary degree. Communications on Pure & Applied Analysis, 2009, 8 (2) : 725-742. doi: 10.3934/cpaa.2009.8.725
[15]

Jaume Llibre, Marco Antonio Teixeira. Regularization of discontinuous vector fields in dimension three. Discrete & Continuous Dynamical Systems, 1997, 3 (2) : 235-241. doi: 10.3934/dcds.1997.3.235
[16]

Jaume Llibre, Ricardo Miranda Martins, Marco Antonio Teixeira. On the birth of minimal sets for perturbed reversible vector fields. Discrete & Continuous Dynamical Systems, 2011, 31 (3) : 763-777. doi: 10.3934/dcds.2011.31.763
[17]

Begoña Alarcón, Víctor Guíñez, Carlos Gutierrez. Hopf bifurcation at infinity for planar vector fields. Discrete & Continuous Dynamical Systems, 2007, 17 (2) : 247-258. doi: 10.3934/dcds.2007.17.247
[18]

Adriana Buică, Jaume Giné, Maite Grau. Essential perturbations of polynomial vector fields with a period annulus. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1073-1095. doi: 10.3934/cpaa.2015.14.1073
[19]

Patrick Bonckaert, P. De Maesschalck. Gevrey and analytic local models for families of vector fields. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 377-400. doi: 10.3934/dcdsb.2008.10.377
[20]

Nataliya Goncharuk, Yury Kudryashov. Families of vector fields with many numerical invariants. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021114

2020 Impact Factor: 0.929

Tools

Metrics

  • PDF downloads (289)
  • HTML views (1942)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences