Bejancu, Aurel Geometry of CR-submanifolds. (English) Zbl 0605.53001 Mathematics and Its Applications (East European Series), 23. Dordrecht etc.: D. Reidel Publishing Company, a member of the Kluwer Academic Publishers Group. XII, 169 p. Dfl. 120.00; $ 49.50; £33.25 (1986). Differential geometry of submanifolds of a Kählerian manifold has been studied extensively in the last three decades. We have three typical classes of submanifolds: holomorphic submanifolds, totally real submanifolds and CR-submanifolds. The study of holomorphic submanifolds of Kählerian manifolds was initiated by E. Calabi [Ann. Math., II. Ser. 58, 1-23 (1953; Zbl 0051.131)] and since then many important results have been obtained [K. Nomizu and B. Smyth, J. Math. Soc. Japan 20, 498-521 (1968; Zbl 0181.501); K. Ogiue, Adv. Math. 13, 73-114 (1974; Zbl 0275.53035)]. The theory of totally real submanifolds was initiated ten years ago [see K. Yano and M. Kon, Anti-invariant submanifolds (1976; Zbl 0349.53055)]. In 1978 the author introduced the notion of CR-submanifolds of a Kählerian manifold [Proc. Am. Math. Soc. 69, 135-142 (1978; Zbl 0368.53040)] as follows: Let N be a Kählerian manifold and let J be the almost complex structure of N. A real submanifold M of N is called a CR- submanifold of N if there is a differentiable distribution D on M such that: i) D is a holomorphic distribution, that is, \(JD_ x=D_ x\) for each \(x\in M\), and ii) the complementary orthogonal distribution \(D^{\perp}\) of D is a totally real distribution, that is, \(JD_ x^{\perp}\subset T_ xM^{\perp}\) for each \(x\in M\), where \(T_ xM^{\perp}\) is the normal space to M at x. The purpose of the book under review is to introduce the reader to the main problems of the geometry of CR-submanifolds and some new structures of submanifolds of several classes of manifolds. Though the research in this field started just a few years ago, here, for the time being, still are a lot of interesting results and some remarkable classification theorems. It is a remarkable fact that the author has made many interesting contributions to the theory of CR-submanifolds. The book is divided into seven chapters. The first chapter deals with the required background material. Chapter II is concerned with CR- submanifolds of almost Hermitian manifolds. The integrability of both of the distribution D and \(D^{\perp}\) are studied. Chapter III deals with some special classes of CR-submanifolds of Kählerian manifolds: umbilical CR-submanifolds, normal CR-submanifolds, CR-products, Sasakian anti-holomorphic submanifolds. The cohomology of CR-submanifolds is also studied according to B.-Y. Chen [Ann. Fac. Sci. Toulouse, V. Sér., Math. 3, 167-172 (1981; Zbl 0478.53046)]. Chapter IV is devoted to the main contributions of D. Blair, B.-Y. Chen, M. Kon, K. Yano and the author to CR-submanifolds of complex space forms. In chapter V the author gives some extensions of CR-structures to other geometrical structures. Many contributions of the author and N. Papaghiuc are included. More results on the contact CR-submanifolds of Sasakian manifolds can be found in the book of K. Yano and M. Kon [CR- submanifolds of Kählerian and Sasakian manifolds (1983; Zbl 0496.53037)]. Chapter VI gives some results on pseudo-conformal mappings on CR-manifolds. In the last chapter, the author proves an application of CR-structures to relativity discovered by R. Penrose [Proc. Symp. Pure Math. 39, Part 1, 401-422 (1983; Zbl 0523.53058)]. The exposition is clear and very carefully organized. The problems exposed in this book give an interesting direction for actual research in differential geometry. This book should be a valuable addition to most libraries. Reviewer: S.Ianus Cited in 40 ReviewsCited in 124 Documents MSC: 53-02 Research exposition (monographs, survey articles) pertaining to differential geometry 53C40 Global submanifolds 53B25 Local submanifolds 53B35 Local differential geometry of Hermitian and Kählerian structures 32V40 Real submanifolds in complex manifolds Keywords:semi-invariant submanifold; Sasakian manifold; Kählerian manifold; CR- submanifolds; complex space forms; CR-structures Citations:Zbl 0376.53034; Zbl 0051.131; Zbl 0181.501; Zbl 0275.53035; Zbl 0349.53055; Zbl 0368.53040; Zbl 0478.53046; Zbl 0496.53037; Zbl 0523.53058 PDFBibTeX XML