Compact generalized Hopf and cosymplectic solvmanifolds and the Heisenberg group \(H(n,1)\)
type: Item
instance of: scholarly article
title: Compact generalized Hopf and cosymplectic solvmanifolds and the Heisenberg group \(H(n,1)\)
zbMATH Open document ID: 0891.53019
DOI: 10.1007/BF02760929
author: Juan Carlos Marrero
author: Edith Padrón
published in: Israel Journal of Mathematics
publication date: 1998-07-19T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 22E25
Mathematics Subject Classification ID: 53C55
zbMATH DE Number: 1123743
zbMATH Keywords: generalized Hopf manifold
zbMATH Keywords: solvmanifold
zbMATH Keywords: locally conformal Kähler manifold
zbMATH Keywords: cosymplectic manifold
reviewed by: Manuel de León
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02760929
OpenAlex ID: W1995355248
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description: scientific article; zbMATH DE number 1123743