Causal Ladder of Finsler Spacetimes with a Cone Killing Vector Field

The correspondence between wind Riemannian structures and spacetimes endowed with a Killing vector field as developed in full generality in Caponio et al. (Memoirs Am Math Soc 300:1501, 2024) is deepened by considering a cone structure endowed with a vector field that preserves the structure (termed cone Killing vector field) and a wind Finslerian structure, introduced in Caponio et al. (Memoirs Am Math Soc 300:1501, 2024) as well. Causality properties of the former are characterized by using metric-type properties of the latter. A particular attention is posed to the case of a cone structure associated with a Finsler-Kropina-type metric, i.e., a field of compact and strongly convex indicatrices that enclose the zero vector in the closure of its bounded interior at each tangent space of the manifold.