On the cohomology algebra of some classes of geometrically formal manifolds

Let $M$ be a closed oriented manifold of dimension $n$ and $\omega$ a closed 1-form on it. We discuss the question whether there exists a Riemannian metric for which $\omega$ is co-closed. For closed 1-forms with nondegenerate zeros the question was answered completely by Calabi in 1969. The goal of this paper is to give an answer in the general case, i.e. not making any assumptions on the zero set of $\omega$.

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 ...

Using the concept of s-formality we are able to extend the bounds of a Theorem of Miller and show that a compact k-connected 4k+3- or 4k+4-manifold with b_{k+1}=1 is formal. We study k connected n-manifolds, n= 4k+3, 4k+4, with a hard Lefschetz-like property and prove that in this case if b_{k+1}=2, then the manifold is formal, while, in 4k+3-dimensions, if b_{k+1}=3 all Massey products vanish. We finish with examples inspired by symplectic geometry and manifolds with special...

We study the construction and classification of weakly Bochner-flat (WBF) metrics (i.e., Kahler metrics with coclosed Bochner tensor) on compact complex manifolds. A Kahler metric is WBF if and only if its `normalized' Ricci form is a hamiltonian 2-form: such 2-forms were introduced and studied in previous papers in the series. It follows that WBF Kahler metrics are extremal. We construct many new examples of WBF metrics on projective bundles and obtain a classification of ...

For a $2n+1$-dimensional compact Sasakian manifold, if $n\ge 2$, we prove that the analytic germ of the variety of representations of the fundamental group at every semi-simple representation is quadratic. To prove this result, we prove the almost-formality of de Rham complex of a Sasakian manifold with values in a semi-simple flat vector bundle. By the almost-formality, we also prove the vanishing theorem on the cup product of the cohomology of semi-simple flat vector bundle...

We construct a simply connected compact manifold which has complex and symplectic structures but does not admit K\"ahler metrics, in the lowest possible dimension where this can happen, that is, dimension 6. Such a manifold is automatically formal and has even odd-degree Betti numbers but it does not satisfy the Lefschetz property for any symplectic form.

We prove that all generalised symmetric spaces of compact simple Lie groups are formal in the sense of Sullivan. Nevertheless, many of them, including all the non-symmetric flag manifolds, do not admit Riemannian metrics for which all products of harmonic forms are harmonic.

The algebra of exterior differential forms on a regular 3-Sasakian 7-manifold is investigated, with special reference to nearly-parallel $G_2$ 3-forms. This is applied to the study of 3-forms invariant under cohomogeneity-one actions by $SO(4)$ on the 7-sphere and on Berger's space $SO(5)/SO(3)$.

In this paper, we prove that the harmonicity of a metric on a pseudo-Riemannian manifold is equivalent to the harmonicity of both its Sasaki (resp. horizontal and complete) lift metric on the tangent bundle and its Sasaki lift metric on the cotangent bundle. Some applications are included.

We survey some recent results and constructions of almost-K\"ahler manifolds whose curvature tensors have certain algebraic symmetries. This is an updated and corrected version of the (to be) published manuscript.