Wind Riemannian spaceforms and Randers metrics of constant flag curvature

Abstract

Recently, wind Riemannian structures (WRS) have been introduced as a generalization of Randers and Kropina metrics. They are constructed from the natural data for Zermelo navigation problem, namely, a Riemannian metric $g_R$ and a vector field $W$ (the wind), where, now, the restriction of mild wind $g_R(W,W)<1$ is dropped. Here, the models of WRS spaceforms of constant flag curvature are determined. Indeed, the celebrated classification of Randers metrics of constant flag curvature by Bao, Robles and Shen, extended to the Kropina case in the works by Yoshikawa, Okubo and Sabau, can be used to obtain the local classification. For the global one, a suitable result on completeness for WRS yields the complete simply connected models. In particular, any of the local models in the Randers classification does admit an extension to a unique model of wind Riemannian structure, even if it cannot be extended as a complete Finslerian manifold. Thus, WRS's emerge as the natural framework for the analysis of Randers spaceforms and, prospectively, wind Finslerian structures would become important for other global problems too. For the sake of completeness, a brief overview about WRS (including a useful link with the conformal geometry of a class of relativistic spacetimes) is also provided.

17 pages, 4 figures

References in corpus (1)

• Stationary Metrics and Optical Zermelo-Randers-Finsler Geometry

Cited by in corpus (7)

• On the definition and examples of cones and Finsler spacetimes
• Wind Finslerian structures: from Zermelo's navigation to the causality of spacetimes
• Foundations of Finsler spacetimes from the Observers' Viewpoint
• Applications of cone structures to the anisotropic rheonomic Huygens' principle
• Huygens' envelope principle in Finsler spaces and analogue gravity
• Anisotropic connections and parallel transport in Finsler spacetimes
• An account on links between Finsler and Lorentz Geometries for Riemannian Geometers