In this paper we first present the real homogeneous complex Finsler metrics ( $$\mathbb {R}$$ -complex Finsler metrics) and make use of them to generalize the Zermelo navigation on Hermitian manifolds. $$\mathbb {R}$$ -complex Randers metrics are obtained by the Zermelo deformation of Hermitian metrics, with variable space-dependent ship’s relative speed under action of weak complex vector fields. Next, we indicate how some properties of a Hermitian metric, e.g. Kahler property, holomorphic sectional curvature behave by the Zermelo deformation in a special holomorphic wind. Lastly, the results are illustrated with some relevant examples.

中文翻译：

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全纯向量场对厄米度量的 Zermelo 变形

在本文中，我们首先介绍了真正的齐次复 Finsler 度量（$$\mathbb {R}$$ -complex Finsler 度量）并利用它们来概括 Hermitian 流形上的 Zermelo 导航。$$\mathbb {R}$$ -complex Randers 度量是通过 Hermitian 度量的 Zermelo 变形获得的，在弱复矢量场的作用下，船舶的相对速度可变。接下来，我们指出 Hermitian 度量的一些性质，例如 Kahler 性质、全纯截面曲率如何在特殊的全纯风中通过 Zermelo 变形表现出来。最后，通过一些相关的例子来说明结果。