Abstract
We study natural Einstein Riemann extensions of torsion-free affine manifolds (M, del). Such a Riemann extension of n-dimensional (M, del) is always a pseudo-Riemannian manifold of signature (n, n).
It is well known that, if the base manifold (M, del) is a torsion-free affine two-manifold with skew-symmetric Ricci tensor, or a flat affine space, we obtain a (globally) Osserman structure on the cotangent bundle T*M over (M, del). If the new base manifold is an arbitrary direct product of the simple affine manifolds described above, we found that the resulting structures on T*M are not Osserman but only "almost Osserman", in the sense that the Jacobi operator has to be restricted from the whole set of space-like unit vectors (or time-like unit vectors, respectively) to a complement of a subset of measure zero.
We also find that the characteristic polynomial of the (restricted) Jacobi operator in the cotangent bundle depends only on the full dimension n of the base manifold, and it is the same as for the flat affine space.