Zermelo Deformation of Hermitian Metrics by Holomorphic Vector Fields
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Zermelo Deformation of Hermitian Metrics by Holomorphic Vector Fields Nicoleta Aldea Abstract. In this paper we first present the real homogeneous complex Finsler metrics (R-complex Finsler metrics) and make use of them to generalize the Zermelo navigation on Hermitian manifolds. 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. K¨ ahler property, holomorphic sectional curvature behave by the Zermelo deformation in a special holomorphic wind. Lastly, the results are illustrated with some relevant examples. Mathematics Subject Classification. 53B40, 53C60, 53B35, 53B50, 32Q15. Keywords. Complex Finsler metric, Hermitian metric, Zermelo navigation, holomorphic deformation.
1. Introduction In Zermelo’s navigation problem, formulated initially by Zermelo (1871–1953) in [28], the objective is to find the paths which minimize travel time of the ship proceeding from a point to another point in presence of perturbing wind W , under assumption the ship sails at constant maximum speed relative to the surrounding sea M . Exploration of this problem leaded to important generalizations and results in Riemann–Finsler geometry. It was shown that the solutions of the Zermelo problem on Riemannian manifold (M, h) are represented by geodesics of a Randers metric (in weak wind) or Kropina metric (in critical wind) [11,12,14,17,23,27]. This subject is still being addressed because of its various applications in the essential theoretical investigations [11–14,17,23,27] as well as in the real world problems [6,15,16,19]. 0123456789().: V,-vol
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Results Math
The Zermelo navigation was also considered on Hermitian manifolds, where the solutions are represented by the metrics of special type in complex Finsler geometry, i.e. complex Randers and complex Kropina metrics. Nevertheless, it was necessary to work out the additional geometric assumptions which come from the complex homogeneity requirement for complex Finsler metrics [4,5]. Without this restriction the solutions of the problem are only real homogeneous, actually R-complex Finsler metrics which have been developed in [2,3,9,21] recently. Therefore, the first purpose of this paper is to describe R-complex Finsler metrics as solutions of Zermelo navigation problem on Hermitian manifolds (M, h), under action of weak wind W and with variable space-dependent ship’s relative speed ||u||h . The second objective is to investigate the holomorphic curvature of a class of R-complex Finsler metrics obtained by Zermelo navigation, that is, as the deformation of some Hermitian metrics by certain holomorphic vector fields. An overview of the paper’s content is pointed out below. In Sect. 2 we summarize some preliminary notions on n-dimensional R-complex Finsler spaces and then we introduce a new class of R-complex Hermitian Randers metrics which are t
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