Structures on manifolds.

*(English)*Zbl 0557.53001
Series in Pure Mathematics, Vol. 3. Singapore: World Scientific. Distr. by John Wiley & Sons Ltd., Chichester. IX, 508 p. £25.30 (1984).

The book under review deals with the most important differential geometrical structures on manifolds and their induced structures on submanifolds.

In the first two chapters the authors recall the background material with respect to Riemannian manifolds and their submanifolds. Then the complex structure, which in fact is one of the most interesting structures on a manifold, is studied in Chapter III. The reader will find results with respect to the geometry of almost Hermitian manifolds, nearly Kählerian manifolds and Kählerian manifolds. In Chapter IV there are studied three important classes of submanifolds: Kählerian submanifolds, anti- invariant submanifolds and CR-submanifolds. The next two chapters are dealing with the contact structures on manifolds and their induced structures on submanifolds. The Chapter VII is devoted to the study of f- structures. Fundamental results with respect to product manifolds and their submanifolds are given in Chapter VIII. Finally, the last chapter is dealing with submersions. By means of exercises at the end of each chapter the reader is informed about some more important results on the topic.

The book is well written and carefully organized. It is an important and welcome contribution to the literature on differential geometry.

In the first two chapters the authors recall the background material with respect to Riemannian manifolds and their submanifolds. Then the complex structure, which in fact is one of the most interesting structures on a manifold, is studied in Chapter III. The reader will find results with respect to the geometry of almost Hermitian manifolds, nearly Kählerian manifolds and Kählerian manifolds. In Chapter IV there are studied three important classes of submanifolds: Kählerian submanifolds, anti- invariant submanifolds and CR-submanifolds. The next two chapters are dealing with the contact structures on manifolds and their induced structures on submanifolds. The Chapter VII is devoted to the study of f- structures. Fundamental results with respect to product manifolds and their submanifolds are given in Chapter VIII. Finally, the last chapter is dealing with submersions. By means of exercises at the end of each chapter the reader is informed about some more important results on the topic.

The book is well written and carefully organized. It is an important and welcome contribution to the literature on differential geometry.

Reviewer: A.Bejancu

##### MSC:

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |

53C15 | General geometric structures on manifolds (almost complex, almost product structures, etc.) |

53C40 | Global submanifolds |