On the cohomology algebra of some classes of geometrically formal manifolds

We investigate harmonic forms of geometrically formal metrics, which are defined as those having the exterior product of any two harmonic forms still harmonic. We prove that a formal Sasakian metric can exist only on a real cohomology sphere and that holomorphic forms of a formal Kähler metric are parallel with respect to the Levi–Civita connection. In the general Riemannian case a formal metric with maximal second Betti number is shown to be flat. Finally we prove that a 6-dimensional manifold with b1 ≠ 1, b2 ⩾ 2 and not having the real cohomology algebra of 𝕋3 × S3 carries a symplectic structure as soon as it admits a formal metric.