On Three Dimensional Affine Szabo Manifolds

Abstract

We consider the cyclic parallel Ricci tensor condition, which is a necessary condition for an affine manifold to be Szabo. We show that, in three dimension, there are affine manifolds which satisfy the cyclic parallel Ricci tensor but are not Szabo. Conversely, it is known that in two dimension, the cyclic parallel Ricci tensor forces the affine manifold to be Szabo. Examples of 3-dimensional affine Szabo manifolds are also given. We prove that an affine surface with skew-symmetric Ricci tensor is affine Szabo. Finally, we give some properties of Riemann extensions defined on the cotangent bundle over an affine Szabo manifold.