On a generalized \(H\)-recurrent Finsler space (Q2869600)

Let \(F^n= (M,g_{ij}(x,y))\) be a Finsler space and \(H^i_{j kh}\) the components of the Berwald curvature tensor. It is called generalized \(H\)-recurrent if NEWLINE\[NEWLINE \mathcal{B}_m H^i_{j kh}=\lambda _m H^i_{j kh} + \mu_m (\delta ^i_j g_{kh}-\delta ^i_k g_{jh}) ,NEWLINE\]NEWLINE where \(\mathcal{B}_m\) is the Berwald covariant differential operator and \(\lambda _m\), \(\mu _m \) are non-null covariant vector fields. The authors prove that if \(F^n\) is a generalized \(H\)-recurrent Finsler space then: the Ricci tensor \(H_{kh}\), the curvature vector \(H_k\) and the scalar curvature \(H\) are nonvanishing; it is Landsberg if \(F^2 H_k = (n-1)Hy_k\) or provided that \(\lambda _m \neq \dot{\partial}\lambda\), where \(\lambda = \lambda _h y^h\) or provided that \(\det H^i_j \neq 0\); it is Riemannian if the \(\lambda_k \) do not depend on the directional variables \(y^j\).