Some geometric correspondences for homothetic navigation

In this paper, we discuss the geodesic and Jacobi field correspondences for homothetic navigation, and then use them to find alternative proofs for some well-known flag curvature and $S$-curvature formulas, and to fully answer the question when a homothetic navigation preserves the local symmetric property in the curvature sense. These geometric correspondences also help us find the local correspondence between isoparametric functions or isoparametric hypersurfaces before and after a homothetic navigation, which generalizes the classification works of Q. He et al. for isoparametric hypersurfaces in Randers space forms and Funk spaces.