##
**Holomorphic curves in symplectic geometry.**
*(English)*
Zbl 0802.53001

Progress in Mathematics (Boston, Mass.). 117. Basel: Birkhäuser. xi, 328 p. (1994).

The articles of this volume will not be indexed individually.

The book is based on lectures given in a CIMPA summer school, held in Sophia-Antipolis, France in July 1992. The aim of the school was to understand in some detail the techniques and results of the famous paper by M. Gromov [Invent. Math. 82, 307-347 (1985; Zbl 0592.53025)]. A short introductory chapter written by the editors contains a very clear description of some problems in symplectic topology and Gromov’s way to solve them. Chapters I–IV are devoted to some basic facts in symplectic geometry, almost complex manifolds, Riemannian geometry, Chern characteristic classes. Then the technical part of the book is started. Chapters V, VI are devoted to analytic and geometric aspects of the theory of pseudo-holomorphic curves, chapter VII gives a proof of Gromov version of the classical Schwarz lemma, which is a main tool in the proof of the Gromov compactness theorem given in chapter VIII. In chapter IX, pseudo-holomorphic curves are shown to appear also in Riemannian geometry. The developed technique is actively exploited in the last chapter X, which gives a panoramic view of Lagrangian submanifolds and related subjects. The book contains contributions of the following specialists, besides the editors: A. Banyaga, P. Gauduchon, F. Labourie, F. Lalonde, Gang liu, D. Mcduff and M.-P. Muller, P. Pansu, L. Polterovich, J. C. Sikorav. The exposition is essentially self-contained, and, in my opinion, the book should be useful to any mathematician, who is interested in symplectic topology, including graduated students.

The book is based on lectures given in a CIMPA summer school, held in Sophia-Antipolis, France in July 1992. The aim of the school was to understand in some detail the techniques and results of the famous paper by M. Gromov [Invent. Math. 82, 307-347 (1985; Zbl 0592.53025)]. A short introductory chapter written by the editors contains a very clear description of some problems in symplectic topology and Gromov’s way to solve them. Chapters I–IV are devoted to some basic facts in symplectic geometry, almost complex manifolds, Riemannian geometry, Chern characteristic classes. Then the technical part of the book is started. Chapters V, VI are devoted to analytic and geometric aspects of the theory of pseudo-holomorphic curves, chapter VII gives a proof of Gromov version of the classical Schwarz lemma, which is a main tool in the proof of the Gromov compactness theorem given in chapter VIII. In chapter IX, pseudo-holomorphic curves are shown to appear also in Riemannian geometry. The developed technique is actively exploited in the last chapter X, which gives a panoramic view of Lagrangian submanifolds and related subjects. The book contains contributions of the following specialists, besides the editors: A. Banyaga, P. Gauduchon, F. Labourie, F. Lalonde, Gang liu, D. Mcduff and M.-P. Muller, P. Pansu, L. Polterovich, J. C. Sikorav. The exposition is essentially self-contained, and, in my opinion, the book should be useful to any mathematician, who is interested in symplectic topology, including graduated students.

Reviewer: A.A.Agrachev

### MSC:

53-06 | Proceedings, conferences, collections, etc. pertaining to differential geometry |

00B25 | Proceedings of conferences of miscellaneous specific interest |

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