The Riemann extension of an affine Osserman connection on 3-dimensional manifold

Abstract

The Riemannian extension of torsion free affine manifolds (M,∇) is an important method to produce pseudo-Riemannian manifolds. It is know that, if the manifold (M,∇) is a torsion-free affine two-dimensional manifold with skew symmetric tensor Ricci, then (M,∇) is affine Osserman manifold . In higher dimensions the skew symmetric of the tensor Ricci is a necessary but not sufficient condition for a affine connection to be Osserman. In this paper we construct affine Osserman connection with Ricci flat but not flat and example of Osserman pseudo-Riemannian metric of signature (3, 3) is exhibited.