Remarks on the product of harmonic forms

TL;DR

This paper investigates conditions under which warped product and Vaisman metrics are formal, linking metric formality to topological restrictions and harmonic form properties.

Contribution

It establishes that warped product metrics are formal only with constant warping functions and provides criteria for Vaisman metrics to be formal.

Findings

Warped product metrics are formal iff the warping function is constant.

Necessary and sufficient conditions for Vaisman metrics to be formal.

Topological obstructions to the existence of formal metrics.

Abstract

A metric is formal if all products of harmonic forms are again harmonic. The existence of a formal metric implies Sullivan formality of the manifold, and hence formal metrics can exist only in presence of a very restricted topology. We show that a warped product metric is formal if and only if the warping function is constant and derive further topological obstructions to the existence of formal metrics. In particular, we determine necessary and sufficicient conditions for a Vaisman metric to be formal.