Construction of an $A$-manifold on a principal torus bundle

Abstract

We construct a new example of an $A$-manifold, i.e. a Riemannian manifold with cyclic-parallel Ricci tensor. This condition can be viewed as a generalization of the Einstein condition. The underlying manifold for our construction is a principal torus bundle over a product of Kähler-Einstein manifolds with fibre a torus of arbitrary dimension.