Given the class of Finsler spaces with Lorentzian signature $(M,L)$ on a manifold $M$ endowed with a timelike vector field $\mathcal{ X}$ satisfying $g_{ (x,y)}(\mathcal{ X},\mathcal{ X})<0$ at any point $(x,y)$ of the slit tangent bundle, a pseudo-Riemannian metric defined on $M$ of signature $n-1$ is associated to the fundamental tensor $g$. Furthermore, an affine, torsion free connection is associated to the Chern connection determined by $L$. The definition of the average connection does not make use of $\mathcal{ X}$. Therefore, there is no direct relation between these two averaged objects.
Categories: math.DG gr-qc math-ph math.MP