The navigation problems and the curvature properties on conic Kropina manifolds

TL;DR

This paper investigates navigation problems on conic Kropina manifolds, showing the resulting metrics are either Randers or Kropina, and explores how curvature properties are affected by this transformation.

Contribution

It characterizes the solutions to navigation problems on conic Kropina manifolds as Randers or Kropina metrics and relates their curvature properties.

Findings

The navigation problem solutions are either Randers or Kropina metrics.

Relationships between curvature properties of original and transformed metrics are established.

Curvature invariants like S-curvature, flag curvature, and Ricci curvature are connected.

Abstract

In this paper, we study navigation problems on conic Kropina manifolds. Let $F(x, y)$ be a conic Kropina metric on an $n$-dimensional manifold $M$ and $V$ be a conformal vector field on $(M, F)$ with $F(x, - V_{x})\leq 1$. Let $\widetilde{F}= \widetilde{F} (x,y)$ be the solution of the navigation problem with navigation data $(F, V)$. We prove that $\widetilde{F}$ must be either a Randers metric or a Kropina metric. Then we establish the relationships between some curvature properties of $F$ and the corresponding properties of the new metric $\widetilde{F}$, which involve S-curvature, flag curvature and Ricci curvature.