We study the behaviour of geodesics on a Randers rotational surface of revolution. The main tool is the extension of Clairaut relation from Riemannian case to the Randers case. Moreover, we consider the embedding problem of this surface in a Minkowski space as a hypersurface. Finally, we study the rays and poles as well as the structure of the cut locus of a Randers rotational surface of revolution of von Mangoldt type.

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The aim of this paper is to determine the structure of the cut locus for a class of surfaces of revolution homeomorphic to a cylinder. Let M denote a cylinder of revolution which admits a reflective symmetry fixing a parallel called the equator of M. It will be proved that the cut locus of a point p of M is a subset of the union of the meridian and the parallel opposite to p respectively, if the Gaussian curvature of M is decreasing on each upper half meridian.