Randers metrics based on deformations by gradient winds

This paper investigates the deformations of Riemannian metrics, in particular Hessian metrics, by Zermelo's navigation under the action of the weak gradient winds. Various descriptions of the resulting Randers metrics are given in relation to other special classes of Finsler metrics, e.g., projectively flat, locally dually flat. We prove that the resulting Randers metric obtained from perturbation by a conformal gradient wind is locally dually flat if and only if the background Riemannian metric is homothetic with the Euclidean metric. The inverse problem answers the question, when a given Randers metric comes from a Hessian metric and a gradient vector field through the Zermelo deformation. Some relevant examples are indicated at the end.