The Geometry of a Randers Rotational Surface with an Arbitrary Direction Wind

Abstract

In the present paper, we study the global behaviour of geodesics of a Randers metric, defined on Finsler surfaces of revolution, obtained as the solution of the Zermelo’s navigation problem. Our wind is not necessarily a Killing field. We apply our findings to the case of the topological cylinder R×S1 and describe in detail the geodesics behaviour, the conjugate and cut loci.

The cut locus of a Randers rotational 2-sphere of revolution

“…To determine the precise structure of the cut locus on a Riemannian or Finsler manifold is not an easy task. The majority of known results concern Riemannian or Randers surfaces of revolution (see [15], [16] for the Riemannian, and [7], [8] for the Randers case).…”

“…The pair (h, W ) will be called the navigation data of the Randers metric F = α + β. Conversely, the Randers metric F = α + β will be called the solution of Zermelo's navigation problem (h, W ). In the case when W is an h-Killing field, provided h is not flat, the geodesics, conjugate points and cut points of the Randers metric F = α + β can be obtained by the translation of the corresponding geodesics, conjugate points and cut points, of the Riemannian metric h by the flow of W , respectively (see [8], [11]). More generally, new Finsler metrics F can be obtained by the rigid translation of the indicatrix of a given Finsler metric F 0 by a vector field W , such that F 0 (−W ) < 1 (see [6], [13]).…”