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 symmetries) is equivalent, in the real analytic case, to the infinite numebr of curvature identities called the odd Ledger conditions. In particular, a Riemannian manifold (M, g) satisfying the first odd Ledger condition L3 is said to be an L3-space.

This definition extends easily to the affine space. Here we investigate the torsion-free affine manifolds (M, /nabla) and their Riemann extensions (T*M, g/bar) as concerns heredity of 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.

Keywords

• Affine connection
• Riemann extension
• D'Atri space
• parallel Ricci tensor