  Recently, a link between Lorentzian and Finslerian Geometries has been
carried out, leading to the notion of wind Riemannian structure (WRS), a
generalization of Finslerian Randers metrics. Here, we further develop this
notion and its applications to spacetimes, by introducing some
characterizations and criteria for the completeness of WRS's.
  As an application, we consider a general class of spacetimes admitting a time
function $t$ generated by the flow of a complete Killing vector field
(generalized standard stationary spacetimes or, more precisely, SSTK ones) and
derive simple criteria ensuring that its slices $t=$ constant are Cauchy.
Moreover, a brief summary on the Finsler/Lorentz link for readers with some
acquaintance in Lorentzian Geometry, plus some simple examples in Mathematical
Relativity, are provided.
