Almost Osserman structures on natural Riemann extensions

We study natural Einstein Riemann extensions of torsion-free affine manifolds (M,∇)(M,∇). Such a Riemann extension of n  -dimensional (M,∇)(M,∇) is always a pseudo-Riemannian manifold of signature (n,n)(n,n). It is well known that, if the base manifold (M,∇)(M,∇) 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⁎MT⁎M over (M,∇)(M,∇). 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⁎MT⁎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.