On formal Riemannian metrics

Formal Riemannian metrics are characterized by the property that all products of harmonic forms are again harmonic. They have been studied over the last ten years and there are still many interesting open conjectures related to geometric formality. The existence of a formal metric implies Sullivan’s formality of the manifold, and hence formal metrics can exist only in presence of a very restricted topology. In this paper we give an overview over the present state of research on geometrically formal manifolds, with emphasis on the recent results obtained by the author together with Liviu Ornea in [11]. We are mainly interested in the topological obstructions to the existence of formal metrics. Moreover, we discuss natural constructions of formal metrics starting from known ones. 1 Motivation, Definitions and Examples A manifold is geometrically formal if it admits a formal Riemannian metric, while a Riemannian metric is called formal if all products of harmonic forms are again harmonic. One of the main motivation for the study of such manifolds is that the existence of a formal metric implies Sullivan’s formality of the manifold. In algebraic topology one wants to read the homotopy type of a space in terms of cohomological data. A precise definition of this property was given