The navigation problems on a class of conic Finsler manifolds
Emerging from the deep understanding of the essential relationships between the Zermelo navigation problems and the indicatrix geometry on Finsler manifolds, the authors study navigation problems on conic Kropina manifolds. For a conic Kropina metric \(F(x,y)\) and a vector field \(V\) with \(F(x,-V_x)\le 1\) on an \(n\)-dimensional manifold \(M\), let \({\tilde F}={\tilde F}(x,y)\) be the solution of the navigation problem with navigation data \((F,V)\). It is proved that \({\tilde F}\) must be either a Randers metric or a Kropina metric. Then, several relationships are established between certain curvature properties of \(F\) and the corresponding properties of \({\tilde F}\), when \(V\) is a conformal vector field on \((M,F)\), which involve \(S\)-curvature, flag curvature and Ricci curvature.
scientific article; zbMATH DE number 7315197
scientific article