Zermelo Deformation of Hermitian Metrics by Holomorphic Vector Fields

Abstract

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.