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**Geometry of CR-submanifolds.**
*(English)*
Zbl 0605.53001

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.

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

### 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 |