Construction of an A-manifold on a principal torus bundle

TL;DR

This paper constructs a new example of an A-manifold, a Riemannian manifold with a cyclic-parallel Ricci tensor, using a principal torus bundle over a Kähler-Einstein base, expanding the class of known manifolds with special Ricci properties.

Contribution

It introduces a novel construction of A-manifolds via principal torus bundles over Kähler-Einstein manifolds, generalizing Einstein manifolds.

Findings

Provides explicit examples of A-manifolds.

Shows the construction works for arbitrary torus dimensions.

Extends the understanding of manifolds with cyclic-parallel Ricci tensors.

Abstract

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