Kropina metrics and Zermelo navigation on Riemannian manifolds

TL;DR

This paper explores Kropina metrics derived from Zermelo's navigation problem on Riemannian manifolds, characterizing those with constant flag curvature and identifying only Euclidean space and odd-dimensional spheres as models.

Contribution

It provides a classification of globally defined Kropina metrics with constant flag curvature, revealing only two model spaces up to local isometry.

Findings

Kropina metrics solve Zermelo's navigation problem on Riemannian manifolds.

Only Euclidean space and odd-dimensional spheres serve as models for constant flag curvature.

The paper characterizes the geometric structure of these special Kropina metrics.

Abstract

The present paper studies globally defined Kropina metrics as solutions of the Zermelo's navigation problem. Moreover, we characterize the Kropina metrics of constant flag curvature showing that up to local isometry, there are only two model spaces of them: the Euclidean space and the odd-dimensional spheres.