Generalized Zermelo navigation on Hermitian manifolds under mild wind

TL;DR

This paper extends the Zermelo navigation problem to Hermitian manifolds with complex perturbations, exploring the geometry of solutions using complex Finsler metrics and providing conditions for projectively flat solutions.

Contribution

It introduces a generalized framework for Zermelo navigation on Hermitian manifolds with complex velocities, analyzing projectively related metrics and flatness conditions.

Findings

Derived conditions for locally projectively flat solutions.

Analyzed the relationship between background Hermitian and Randers metrics.

Provided examples illustrating the theoretical results.

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 $||u(z)||_h\leq1$ 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.