Wind Riemannian spaceforms and Randers metrics of constant flag curvature

TL;DR

This paper classifies wind Riemannian spaceforms of constant flag curvature, extending known results for Randers and Kropina metrics, and explores their global and local geometric properties within the framework of wind Riemannian structures.

Contribution

It extends the classification of Randers and Kropina metrics to wind Riemannian structures, providing local and global models of constant flag curvature.

Findings

Local models extend to unique global models of wind Riemannian structures.

WRS models include all Randers spaceforms and can be globally complete.

Wind Riemannian structures serve as a natural framework for analyzing Randers spaceforms.

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.