Coeffective and de Rham cohomologies on almost contact manifolds
The coeffective cohomology for symplectic manifolds was introduced by \textit{T. Bouché} [Bull. Sci. Math. 114, No. 2, 115-122 (1990; Zbl 0714.58001)] and it was studied by the authors [Math. Z. 226, 11-23 (1997; Zbl 0886.58002) and J. Geom. Phys. 491 (1998)]. In the present paper, the authors consider the coeffective cohomology for almost contact manifolds with closed fundamental 2-form. The interest of the authors is to obtain topological and/or geometrical information of the involved structures by means of the coeffective cohomology. First, the coeffective cohomology is related with the de Rham cohomology by means of a long exact sequence in cohomology. This relation permits to prove a finiteness theorem for this cohomology, a computation theorem for some kinds of homogeneous manifolds (nilmanifolds and completely solvable manifolds) and that the coeffective numbers are bounded by topological bounds (depending on the Betti numbers). The lower bound is attained for compact cosymplectic manifolds, and the upper one for contact and (non-compact) exact almost cosymplectic manifolds. Moreover, different examples are shown. Finally, the authors study the behaviour of the coeffective cohomology under deformations of the almost contact structure; for which it is proved a weak version of the Moser stability theorem for almost cosymplectic manifolds.
scientific article; zbMATH DE number 1186365
scientific article