SOME PROPERTIES OF ZERMELO NAVIGATION IN PSEUDO-FINSLER METRICS UNDER AN ARBITRARY WIND

We generalize the notion of Zermelo navigation to arbitrary pseudo-Finsler metrics possibly defined in conic subsets. The translation of a pseudo-Finsler metric F is a new pseudo-Finsler metric whose indicatrix is the translation of the indicatrix of F by a vector field W at each point, where W is an arbitrary vector field, without the classical restriction F(-W) < 1. Then we show that the Matsumoto tensor of a pseudo-Finsler metric is equal to zero if and only if it is the translation of a semi-Riemannian metric, and when W is homothetic, we give a description of the geodesic flow of the translation and we prove that the flag curvature of the translation coincides with the one of the original one up to the addition of a non-positive constant. We give a proof of the latter by exploiting the fanning curve approach to the flag curvature. These results allow us to extend all the Randers spaceforms classified in [5] to geodesically complete conic Finsler manifolds with constant flag curvature.