The navigation problems on a class of conic Finsler manifolds

Abstract

Based on our deep understanding on the essential relationships between the Zermelo navigation problems and the geometries of indicatrix on Finsler manifolds, we study navigation problems on conic Kropina manifolds. For a conic Kropina metric F(x,y) and a vector field V with F(x,−Vx)≤1 on an n-dimensional manifold M, let F˜=F˜(x,y) be the solution of the navigation problem with navigation data (F,V). We prove that 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 F˜ when V is a conformal vector field on (M,F), which involve S-curvature, flag curvature and Ricci curvature.