Locally conformal Kähler geometry
type: Item
instance of: scholarly article
title: Locally conformal Kähler geometry
zbMATH Open document ID: 0887.53001
author: Sorin Dragomir
author: Liviu Ornea
published in: Progress in Mathematics
publication date: 1998-01-05T00:00:00Z
review text: A Hermitian manifold \((M^{2n},J,g)\) of complex dimension \(n\) is called a locally conformal Kähler (l.c.K.) manifold if the metric \(g\) is conformal to some local Kählerian metric in the neighborhood of each point of \(M^{2n}\). Nowadays, complex geometry deals primarily with Kähler manifolds. The book under review is the first book to treat the geometry of locally conformal Kähler manifolds in a systematic way. The first eleven chapters of this book report on the main achievements in the theory of l.c.K. manifolds. In the first two chapters, the authors present basic results on l.c.K. manifolds, including all known results on Vaisman's conjecture. Chapter 3 provides many nice examples of l.c.K. manifolds. Chapters 4 and 5 discuss generalized Hopf manifolds, i.e., locally conformally Kähler manifolds with parallel Lee form. In Chapters 6 and 7, the authors present the geometry of Vaisman manifolds and generalized Vaisman manifolds. In Chapter 8 they describe the l.c.K. surfaces among the Hermitian surfaces. In particular, many characterizations of Vaisman surfaces are presented in this chapter. Results on holomorphic maps between l.c.K. manifolds are given in Chapter 9. In Chapter 10 the authors present results of Marrero and Rocha on submersions from a l.c.K. manifold. The geometry and topology of locally conformal hyper-Kähler manifolds are given in Chapter 11. The last six chapters present the main achievements in the theory of submanifolds in l.c.K. manifolds. Basic results on submanifolds are included in Chapter 12. Totally umbilical submanifolds and extrinsic spheres of l.c.K. manifolds are discussed in Chapter 13. The geometry of real hypersurfaces of an l.c.K. manifold is discussed in detail in Chapter 14. Chapter 15 investigates complex submanifolds of l.c.K. manifolds. A Frankel type theorem for complex submanifolds in a l.c.K. manifold is also presented in this chapter. The main purpose of Chapter 16 is to present a classification theorem for compact …
Mathematics Subject Classification ID: 53-02
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 53C56
zbMATH DE Number: 1099336
zbMATH Keywords: locally conformal Kähler manifold
zbMATH Keywords: Kähler manifold
zbMATH Keywords: \(CR\)-submanifold
zbMATH Keywords: extrinsic sphere
zbMATH Keywords: Vaisman's conjecture
zbMATH Keywords: Chen class
zbMATH Keywords: locally conformal hyper-Kähler manifold
reviewed by: Bang-Yen Chen
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1099336