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arXiv · 1611.03742

Generalized Zermelo navigation on Hermitian manifolds under mild wind

Nicoleta Aldea·Piotr Kopacz
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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.

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Nicoleta Aldea, Piotr Kopacz. 2016-11-11. Generalized Zermelo navigation on Hermitian manifolds under mild wind. https://doi.org/10.1016/j.difgeo.2017.05.007

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