Nonlinear connections in a Finsler space (Q1120837)

On a Finsler space \(F_ n\) connections \(\Gamma^{1}{}^ i_ k(x,X)\) and \(\Gamma^{2}_{jk}(x,Y)\) nonlinear in the vectors \(X^ i(x)\) resp. \(Y_ j(x)\) are considered and the torsion tensors \(\Omega^{1}_ j{}^ i_ k:=\Gamma^{1}_ j{}^ i_ k-\Gamma^{1}_ k{}^ i_ j,\) \(\Gamma^{1}_ j{}^ i_ k\equiv \frac{\partial}{\partial X^ j}\Gamma^{1}{}^ i_ k\); \(\Omega^{2}_ j{}^ i_ h:=\Gamma^{2}_ j{}^ i_ h-\Gamma^{2}_ h{}^ i_ j,\) \(\Gamma^{2}_ j{}^ i_ h\equiv \frac{\partial}{\partial Y_ i}\Gamma^{2}_{j\;k}\) are studied. It is shown that if geodesics are auto-parallel curves, then \(\Omega^{2}_ j{}^ i_ k=0.\) Also ex-central connections and semi-symmetric connections are investigated under the condition that geodesics are auto-parallel curves.