Topology of cosymplectic manifolds
A \(2n\)-dimensional compact Kähler manifold \(M^{2n}\) has the following five topological properties: 1) The Betti numbers \(b_{2i}\), \(1\leq i\leq n\) are non-zero. 2) The Betti numbers \(b_{2i- 1}\) are even. 3) The dimension of the space of effective harmonic \(p\)-forms is \(b_p- b_{p- 2}\), for \(p\leq n+ 1\), and consequently \(b_{p- 2}\leq b_p\). 4) \(M^{2n}\) has the strong Lefschetz property, viz. the mapping \(L^{n- p}\) of harmonic \(p\)-forms to harmonic \((2n- p)\)-forms induced by \(L\alpha= \alpha\wedge \Omega\) is an isomorphism, \(\Omega\) being the fundamental 2-form of the Kähler structure. 5) The minimal model of \(M^{2n}\) is formal (so in particular all Massey products of \(M^{2n}\) vanish). The purpose of this paper is to obtain similar results for a compact cosymplectic manifold \(M^{2n+ 1}\). In the reviewer's paper with \textit{S. I. Goldberg} [J. Differ. Geom. 1, 347-354 (1967; Zbl 0163.43902)] it was shown that: 1) The Betti numbers \(b_i\) are non-zero for all \(i\). 2) The dimension of the space of effective harmonic \(p\)-forms is \(b_p- b_{p- 2}\), for \(p\leq n+ 1\), and consequently \(b_{p- 2}\leq b_p\) (an error in the proof is corrected by a stronger result in the present paper). The main results of the present paper are: 1) On a compact cosymplectic manifold \(M^{2n+ 1}\), \(b_0\leq b_1\leq\cdots\leq b_n= b_{n+ 1}\) and \(b_{n+ 1}\geq b_{n+ 2}\geq\cdots \geq b_{2n+ 1}\). 2) On a compact cosymplectic manifold \(M^{2n+ 1}\) the differences \(b_{2p+ 1}- b_{2p}\) with \(0\leq p\leq n\) are even and so in particular the first Betti number of \(M^{2n+ 1}\) is odd. 3) A strong Lefschetz property. 4) A compact cosymplectic manifold \(M^{2n+ 1}\) is formal. The notion of cosymplectic used in this paper is the stronger one, viz. a normal almost contact metric manifold for which both the fundamental 1-form and fundamental 2-form are closed. The other notion of a cosymplectic manifold in the literature is a manifold \(M^{2n+ 1}\) admitting a closed 1-form \(\eta\) and a closed 2-form \(\Phi\) such that \(\eta\wedge \Phi^n\neq 0\); in particular the role of normality is not used. A cosymplectic manifold is locally the product of a Kähler manifold and a one-dimensional manifold. Also for a complete, but not compact, simply connected cosymplectic manifold the product structure is global (see the above paper of S. I. Goldberg and the reviewer). The paper closes with a nice example of a compact cosymplectic manifold which is not a global product.
scientific article; zbMATH DE number 811435