Coeffective and de Rham cohomologies on almost contact manifolds
type: Item
instance of: scholarly article
title: Coeffective and de Rham cohomologies on almost contact manifolds
zbMATH Open document ID: 0940.53022
author: Manuel de León
author: Marisa Fernández
author: Raúl Ibañez
published in: Differential Geometry and its Applications
publication date: 1998-09-08T00:00:00Z
review text: 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.
reviewed by: Manuel de León
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 57R19
Mathematics Subject Classification ID: 58A12
zbMATH DE Number: 1186365
zbMATH Keywords: almost cosymplectic manifolds
zbMATH Keywords: contact manifolds
zbMATH Keywords: coeffective cohomology
zbMATH Keywords: nilmanifolds
zbMATH Keywords: deformations
MaRDI profile type: Publication
cites work: Le théorème de réduction de Marsden-Weinstein en géométrie cosymplectique et de contact. (The Marsden-Weinstein reduction theorem in cosymplectic and contact geometry)
cites work: Flows on Homogeneous Spaces. (AM-53)
cites work: Contact manifolds in Riemannian geometry
cites work: Topology of almost contact manifolds
cites work: Q3200265
cites work: Gradient vector fields on cosymplectic manifolds
cites work: Q4725286
cites work: Topology of cosymplectic manifolds
cites work: Coeffective cohomology on almost cosymplectic manifolds
cites work: A canonical differential complex for Jacobi manifolds
cites work: A Nomizu's theorem for the coeffective cohomology
cites work: Coeffective and de Rham cohomologies of symplectic manifolds
cites work: Q4150360
cites work: Q3279826
cites work: Harmonic cohomology classes of almost cosymplectic manifolds
cites work: Q3203663
cites work: On the Volume Elements on a Manifold
cites work: On the cohomology of compact homogeneous spaces of nilpotent Lie groups
cites work: Q3750580
cites work: Harmonic forms and Betti numbers of certain contact Riemannian manifolds
cites work: Q3874497
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description: scientific article; zbMATH DE number 1186365