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Locally conformal Kähler geometry. (English) Zbl 0887.53001
Progress in Mathematics (Boston, Mass.). 155. Boston, MA: Birkhäuser. xi, 327 p. (1998).
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 minimal $$CR$$-submanifolds in a complex Hopf manifold using the integral formula method. In the last chapter, the authors present results on stability, the canonical structures, Chen’s class on $$CR$$-submanifolds, geometric symmetries, and submersed $$CR$$-submanifolds of l.c.K. manifolds.
The authors did a very good job of arranging all important results on l.c.K. manifolds in one place. Since the original material is widely scattered in the literature, readers interested in l.c.K. manifolds will find it enormously helpful to have the results assembled in one place with a unified exposition. This book should be a valuable addition to all research libraries.

##### MSC:
 53-02 Research exposition (monographs, survey articles) pertaining to differential geometry 53C55 Global differential geometry of Hermitian and Kählerian manifolds 53C56 Other complex differential geometry