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Proceedings of the London Mathematical Society Advance Access originally published online on November 4, 2008
Proceedings of the London Mathematical Society 2009 98(3):607-630; doi:10.1112/plms/pdn047
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© 2008 London Mathematical Society

On the cohomology algebra of some classes of geometrically formal manifolds

J.-F. Grosjean

Institut Élie Cartan
Université H. Poincaré
Nancy I, B.P.239
F-54506 Vandoeuvre-Lès-Nancy
France

P.-A. Nagy

Department of Mathematics
University of Auckland
Private Bag 92019
Auckland
New Zealand
nagy@math.auckland.ac.nz

Received 21 August 2006.

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 x S3 carries a symplectic structure as soon as it admits a formal metric.

2000 Mathematics Subject Classification 53C12, 53C24, 53C55.


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