The Riemann extensions with cyclic parallel Ricci tensor

Abstract

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.