Almost Osserman structures on natural Riemann extensions
In the present paper, the authors study natural Einstein-Riemann extensions of torsion-free affine manifolds. The Riemann extension of such a manifold of dimension \(n\) is a pseudo-Riemannian manifold of signature \((n,n)\). It is proven that the natural Einstein-Riemann extension of a direct product of any number of non-flat torsion-free affine two-dimensional manifolds with skew-symmetric Ricci tensors and of a flat affine space is an almost Osserman manifold 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. It is also shown 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 remains the same as for the flat affine space.
scientific article; zbMATH DE number 6141628