A-manifolds on a principal torus bundle over an almost Hodge A-manifold base

Grzegorz Zborowski

doi:10.17951/a.2015.69.1.109

Abstract

An A-manifold is a manifold whose Ricci tensor is cyclic-parallel, equivalently it satisfies ∇<sub>X</sub> Ric(X, X) = 0. This condition generalizes the Einstein condition. We construct new examples of A-manifolds on r-torus bundles over a base which is a product of almost Hodge A-manifolds.

Highlights

One of the most extensively studied objects in mathematics and physics are Einstein manifolds, i.e. manifolds whose Ricci tensor is a constant multiple of the metric tensor
Gray defined a condition which generalizes the concept of an Einsten manifold. This condition states that the Ricci tensor Ric of the Riemannian manifold (M, g) is cyclic parallel, i.e
Over every almost Hodge A-manifold with J-invariant Ricci tensor we can construct a Riemannian metric such that the total space of the bundle is an A-manifold

Summary

Introduction

One of the most extensively studied objects in mathematics and physics are Einstein manifolds (see for example [1]), i.e. manifolds whose Ricci tensor is a constant multiple of the metric tensor. Gray defined a condition which generalizes the concept of an Einsten manifold This condition states that the Ricci tensor Ric of the Riemannian manifold (M, g) is cyclic parallel, i.e. A-manifold, cyclic parallel Ricci, torus bundle, Einstein-like manifold, Killing tensor. Over every almost Hodge A-manifold with J-invariant Ricci tensor we can construct a Riemannian metric such that the total space of the bundle is an A-manifold. Of particular interest in this work is a situation when the Ricci tensor of the metric g is a Killing tensor We call such a manifold an A-manifold. In the more general situation, when the Ricci tensor is a conformal tensor we call (M, g) a AC⊥-manifold. Let X be any vector field and φ, ψ conformal Killing p-forms.

The same is valid for ψ with

As a result we have r

Jk Ei