The Riemann extensions with cyclic parallel Ricci tensor

The property of being a D'Atri space (i.e., a Riemannian manifold with volume-preserving geodesic symmetries) is equivalent, in the real analytic case, to the infinite number of curvature identities called the odd Ledger conditions. In particular, a Riemannian manifold satisfying the first odd Ledger condition L₃ is said to be an L₃-space. This definition extends easily to the affine case. Here we investigate the torsion-free affine manifolds and their Riemann extensions as concerns heredity of the condition L₃. We also incorporate a short survey of the previous results in this direction, including also the topic of D'Atri spaces.