Remarks on the product of harmonic forms

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 the 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 the necessary and sufficient conditions for a Vaisman metric to be formal. IntroductionA fundamental problem in algebraic topology is the reading of the homotopy type of a space in terms of cohomological data. A precise definition of this property was given by Sullivan [1977] and called formality. As concerns manifolds, it is known, for example, that all compact Riemannian symmetric spaces and all compact Kähler manifolds are formal. For a recent survey of topological formality, see [Papadima and Suciu 2009].Sullivan also observed that if a compact manifold admits a metric such that the wedge product of any two harmonic forms is again harmonic, then, by Hodge theory, the manifold is formal. This motivated the following definition: In particular, the length of any harmonic form with respect to a formal metric is (pointwise) constant. This larger class of metrics having all harmonic (one-)forms of constant length naturally appears in other geometric contexts, for instance in the study of certain systolic inequalities, and has been investigated in [Nagy 2006;Nagy and Vernicos 2004].Classical examples of geometrically formal manifolds are compact symmetric spaces. In [Kotschick and Terzić 2003;2011] more general examples are provided, Both authors are partially supported by CNCSIS grant PNII IDEI contract 529/2009. M. Pilca also acknowledges partial support from SFB/TR 12. MSC2000: primary 53C25; secondary 53C55, 58A14.

Vaisman nilmanifolds

“…Similarly, a nilmanifold quotient of is not formal and has triple Massey products. It would be interesting to understand to which extent the existence of a Vaisman structure on a compact manifold influences the vanishing of higher order Massey products (see also ).…”

On formal Riemannian metrics

“…(ii) Warped Products. In this case, also by a direct but rather lenghty computation, we have shown the following (for the proof we refer to [11]): Theorem 1.5 ( [11]). Let (B n , g B ) and (F m , g F ) be two compact Riemannian manifolds with formal metrics.…”

Locally conformal symplectic nilmanifolds with no locally conformal Kähler metrics

“…Can one, in some sense to be made precise, "bound" the non-formality of a compact lcK manifold (for instance, in terms of non-zero Massey products)? The study of geometric formality in the context of Vaisman metrics was tackled in [31].…”