The second order recurrent areal spaces of the submetric class (Q1074128)

Let \(A_ n^{(m)}\) (1\(\leq m\leq n-1)\) be an areal space of the submetric class and let \(\Gamma^*{}^ i_{jk}(x,p)\) be the components of the symmetric connection established by \textit{A. Kawaguchi} and \textit{K. Tandai} [Tensor, New Ser. 2, 47-58 (1952; Zbl 0049.236)]. In this paper the curvature tensors \(R^ h_{ijk}\) and \(K^ h_{ijk}\) are studied under the assumption of recurrence of second order. Various relations between such recurrence properties are pointed out. The properties of the recurrence tensor are investigated in more detail in the case of an affinely connected \(A_ n^{(m)}\), i.e. \(\Gamma^*{}^ i_{jk}\) is independent of p.