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A natural Riemann extension is a natural lift of a manifold with a symmetric affine connection to its cotangent bundle. The corresponding structure on the cotangent bundle is a pseudo-Riemannian metric. The classical Riemann extension has been studied by many authors. The broader (two-parameter) family of all natural Riemann extensions was found by the second author in 1987. We prove the equivariance property for the natural Riemann extensions. We also prove some theorems for Ricci curvature and scalar curvature.

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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 (M,g) 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 (M,∇) and their Riemann extensions (T*M,g¯) 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.

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