Average of geometric structures in Finsler spaces with Lorentzian signature

Given the class of Finsler spaces with Lorentzian signature [Formula: see text] on a manifold endowed timelike vector field satisfying at any point slit tangent bundle, pseudo-Riemannian metric defined is associated to fundamental tensor text]. Furthermore, an affine, torsion free connection Chern determined by The definition average does not make use Therefore, there no direct relation between these two averaged objects.