Superwind and navigation of least time on Riemannian manifolds

Abstract

In this work, minimum-time navigation on Riemannian manifolds is analyzed by means of Finsler geometry. The introduced notion of a superwind being anisotropic, reducible, and unknown a priori encompasses a wind in the spirit of Zermelo's navigation, and a gravitational wind in the sense of a slippery slope model, including Matsumoto's navigation on a mountain slope as the particular cases. The considered generalization also deploys a non-uniform slope, where the counterbalance to the transverse impact of superwind causing a lateral drift varies over space. This requires in particular an extension of the existing theory of general $(α, β)$-metrics, which is presented in our study. The solution of the navigation problem is given by a new Finsler metric and the time-minimizing geodesics, which are determined. Moreover, the thorough analysis establishes the necessary and sufficient conditions for strong convexity under which the resultant velocity defines a Finsler metric. For completeness, a variety of comprehensive and comparative examples are explored in dimension two.