Generalized Zermelo navigation on Hermitian manifolds under mild wind

Abstract

We generalize and study the Zermelo navigation problem on Hermitian manifolds in the presence of a perturbation $W$ determined by a mild complex velocity vector field $||W(z)|{|}_h<||u(z)|{|}_h$, with application of complex Finsler metric of complex Randers type. By admitting space-dependence of ship's relative speed $||u(z)|{|}_h\le 1$ we discuss the projectively related complex Finsler metrics, the geodesics corresponding to the solutions of Zermelo's problem, in particular the conformal case and the connections between the corresponding background Hermitian metric $h$, new Hermitian metric $a$ and resulting complex Randers metric $F$. Moreover, we present some necessary and sufficient conditions for the obtained locally projectively flat solutions. Our findings are also illustrated with several examples.

19 pages

References in corpus (2)

• Application of planar Randers geodesics with river-type perturbation in search models
• A note on time-optimal paths on perturbed spheroid

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• A note on generalization of Zermelo navigation problem on Riemannian manifolds with strong perturbation