Generalized-Finslerian connection coefficients for the static spherically symmetric space (Q1332269)

Assume that a symmetric nondegenerate tensor \(r_{ij} (x^ k)\) of type (0,2) is given on a differentiable \(n\)-manifold and consider the case where the space \((M, r_{ij})\) is static spherically symmetric, which implies that coordinates \(x^ i = (x^ 0, x^ a)\) can be introduced on \(M\), so that \(r_{00} = r_{00}(r)\), \(r_{ab} = -W(r) \delta_{ab}\), \(r_{0a} = 0\), where \(r = (\delta_{ab} x^ a x^ b)^{1/2}\), \(r_{00} > 0\), \(W \neq 0\), and \(\delta\) stands for the Kronecker symbol; the indices \(a, b, \dots\) run from 1 to \(n - 1\). Further we consider two positive scalars \(A_ 1\) and \(A_ 2\) on \(TM\), assuming that these are functions of \(r\) and \(q = -r_{ab} y^ a y^ b[r_{00} (y^ 0)^ 2]^{-1}\), and construct a generalized Finslerian metric tensor \(a_{ij}\) by putting \(a_{00} = A_ 1 r_{00}\), \(a_{ab} = A_ 2 r_{ab}\), \(a_{0a} = 0\). Under these conditions we call \((M, a_{ij})\) a static spherically symmetric generalized-Finslerian space. The purpose of the present paper is to determine a metrical and torsionless connection.