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

Some criteria for Wind Riemannian completeness and existence of Cauchy hypersurfaces