Generalized Hopf manifolds, locally conformal Kaehler structures and real hypersurfaces
type: Item
instance of: scholarly article
title: Generalized Hopf manifolds, locally conformal Kaehler structures and real hypersurfaces
zbMATH Open document ID: 0751.53020
author: Sorin Dragomir
published in: Kodai Mathematical Journal
publication date: 1992-09-26T00:00:00Z
review text: First the author studies a certain action of \(\mathbb{Z}\) on the product of the punctured open unit disk in \(\mathbb{C}\) and some Brieskorn manifold. The resulting quotient space is proved to be a complex manifold and, in dimension two, a complex analytic family of compact complex surfaces. Then he turns to the complex Hopf manifold \(H^ n_ a\), which is the quotient space of \(\mathbb{C}^ n\setminus\{0\}\) by the group of transformations of \(\mathbb{C}^ n\setminus\{0\}\) generated by \(z\mapsto az\), \(a\in\mathbb{C}\), \(0<| a| < 1\), \(n>1\). It is known, that \(H^ n_ a\) admits no globally defined Kähler metric, and the author proves that this also holds for the complex sphere \(\{z\in H^ n_ a\mid z^ 2_ 1+\dots+z^ 2_ n=0\}\) in \(H^ n_ a\), provided that \(a\in\mathbb{R}\) and \(n>4\). Next he shows that the canonical projection \(H^ n_ a\to \mathbb{C} P^{n-1}\) becomes a Riemannian submersion with minimal fibers, when endowing \(H^ n_ a\) with the Boothby metric and \(\mathbb{C} P^{n-1}\) with the Fubini-Study metric. A \(P_ 0K\)-manifold is a locally conformal Kähler manifold with parallel Lee form whose local Kähler metrics are flat. According to \textit{I. Vaisman} [Rend. Mat., VI. Ser. 12, 263-264 (1979; Zbl 0447.53032)] any \(P_ 0K\)-manifold is locally analytically homothetic to some \(H^ n_ a\) with the Boothby metric. In the latter part of the paper the author derives several results about real hypersurfaces with two principal curvatures, quasi-Einstein real hypersurfaces and homogeneous real hypersurfaces in \(P_ 0K\)-manifolds. Finally he proves that every totally umbilical submanifold in \(H^ n_ a\) tangent to the Lee field is either totally geodesic or an extrinsic sphere.
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 53C40
zbMATH DE Number: 55872
zbMATH Keywords: real hypersurfaces
zbMATH Keywords: Brieskorn manifold
zbMATH Keywords: complex Hopf manifold
zbMATH Keywords: Riemannian submersion
zbMATH Keywords: Boothby metric
zbMATH Keywords: \(P_ 0K\)-manifold
zbMATH Keywords: Lee form
zbMATH Keywords: totally umbilical submanifold
reviewed by: Jürgen Berndt
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138039462
OpenAlex ID: W2084565725
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DOI: 10.2996/KMJ/1138039462