Compact generalized Hopf and cosymplectic solvmanifolds and the Heisenberg group \(H(n,1)\)
An almost Hermitian manifold is said to be locally conformal Kähler (l.c.K.) if its metric is conformally related to a Kähler metric in some neighbourhood of every point of \(M\). For an l.c.K. manifold \(M\), the Lee 1-form \(\omega\) is closed. If, moreover, \(\omega\) is parallel and \(\omega\neq 0\), then \(M\) is called a generalized Hopf (g.H.) manifold. A compact g.H. manifold cannot admit Kähler structures. On the other hand, an almost contact metric structure \((\varphi, \xi,\eta,h)\) on a manifold \(N\) with fundamental 2 form \(\Phi\) is said to be: Sasakian if \([\varphi,\varphi] +2d\eta \otimes\xi =0\) and \(d\eta= \Phi\); cosymplectic if \([\varphi, \varphi] =0\), \(d\eta =0\) and \(d\Phi=0\). Sasakian and cosymplectic manifolds are the natural odd-dimensional counterparts to Kähler manifolds. The product of a compact Sasakian manifold with the circle \(S^1\) is a compact g.H. manifold. In particular, the products \(S^{2n-1} \times S^1\) and \(N(n-1,1)\times S^1\) are compact g.H. manifolds, where \(S^{2n-1}\) is the \((2n-1)\)-dimensional unit sphere, and \(N(n-1,1)\) is a compact quotient of the generalized Heisenberg group \(H(n,1)\) by a discrete subgroup. In this paper, the authors prove the following Theorem 2.1: Let \((N,\varphi, \xi,\eta,h)\) be a cosymplectic manifold with integral fundamental 2-form \(\Phi\) and let \(\pi:M\to N\) be the principal circle bundle over \(N\) corresponding to \([\Phi]\in H^2(N,\mathbb{Z})\). Then \(M\) is a g.H. manifold. Using this result, they obtain new and interesting examples of compact generalized Hopf manifolds. Moreover, they give nice descriptions of these examples as compact solvmanifolds and as suspensions with fibre a compact quotient of the group \(H(n,1)\) by a discrete subgroup.
scientific article; zbMATH DE number 1123743
scientific article