Normalized semi parallel Ch-vector field in special Finsler spaces (Q1283647)

A normalized vector field \(X\) in a Finsler space is said to be a normalized semi-parallel Ch-vector field (nspch), if (1) \(\nabla^v X=0\), (2) \(LC_j{}^i{}_k X_i= \alpha h_{jk}+ \beta L^2 C_j C_k\), (3) \(\nabla^h{}_j X_i= \rho(g_{ij}- X_i X_j)\), in the Cartan connection, where \(\alpha,\beta\) and \(\rho\) are scalar functions. \(X_i\) together with \(\alpha\), \(\beta\) and \(\rho\) are assumed to be positively homogeneous functions of degree zero in \(y^i\). If an \(S\)-3 like space admits (nspch), then the \(v\)-curvature tensor \(S\) vanishes. If the space of scalar curvature \(R\) admits an (nspch) \(X\), then \(X\) is written as \(X_i= \lambda y_i+\mu\partial_i\rho\), provided \(L^2(\rho^2+R)+ X_0(\partial_i\rho) y^i\neq 0\).