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On the cohomology algebra of some classes of geometrically formal manifolds
Author(s) -
Grosjean J.F.,
Nagy P.A.
Publication year - 2009
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/pdn047
Subject(s) - mathematics , pure mathematics , betti number , cohomology , metric (unit) , connection (principal bundle) , algebra over a field , manifold (fluid mechanics) , symplectic geometry , holomorphic function , harmonic , geometry , mechanical engineering , operations management , physics , quantum mechanics , engineering , economics
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 b 1 ≠ 1, b 2 ⩾ 2 and not having the real cohomology algebra of 3 × S 3 carries a symplectic structure as soon as it admits a formal metric.
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