# American Institute of Mathematical Sciences

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

## 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

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.

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
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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 $\xi + v$