The school, the book This book is based on lectures given by the authors of the various chapters in a three week long CIMPA summer school, held in Sophia-Antipolis (near Nice) in July 1992. The first week was devoted to the basics of symplectic and Riemannian geometry (Banyaga, Audin, Lafontaine, Gauduchon), the second was the technical one (Pansu, Muller, Duval, Lalonde and Sikorav). The final week saw the conclusion ofthe school (mainly McDuffand Polterovich, with complementary lectures by Lafontaine, Audin and Sikorav). Globally, the chapters here reflect what happened there. Locally, we have tried to reorganise some ofthe material to make the book more coherent. Hence, for instance, the collective (Audin, Lalonde, Polterovich) chapter on Lagrangian submanifolds and the appendices added to some of the chapters. Duval was not able to write up his lectures, so that genuine complex analysis will not appear in the book, although it is a very current tool in symplectic and contact geometry (and conversely). Hamiltonian systems and variational methods were the subject of some of Sikorav's talks, which he also was not able to write up. On the other hand, F. Labourie, who could not be at the school, wrote a chapter on pseudo-holomorphic curves in Riemannian geometry.

Klappentext

This book is devoted to pseudo-holomorphic curve methods in symplectic geometry. It contains an introduction to symplectic geometry and relevant techniques of Riemannian geometry, proofs of Gromov's compactness theorem, an investigation of local properties of holomorphic curves, including positivity of intersections, and applications to Lagrangian embeddings problems. The chapters are based on a series of lectures given previously by the authors M. Audin, A. Banyaga, P. Gauduchon, F. Labourie, J. Lafontaine, F. Lalonde, Gang Liu, D. McDuff, M.-P. Muller, P. Pansu, L. Polterovich, J.C. Sikorav. In an attempt to make this book accessible also to graduate students, the authors provide the necessary examples and techniques needed to understand the applications of the theory. The exposition is essentially self-contained and includes numerous exercises.

Inhalt
Introduction: Applications of pseudo-holomorphic curves to symplectic topology.- 1 Examples of problems and results in symplectic topology.- 2 Pseudo-holomorphic curves in almost complex manifolds.- 3 Proofs of the symplectic rigidity results.- 4 What is in the book and what is not.- 1: Basic symplectic geometry.- I An introduction to symplectic geometry.- II Symplectic and almost complex manifolds.- 2: Riemannian geometry and linear connections.- III Some relevant Riemannian geometry.- IV Connexions linéaires, classes de Chern, théorème de Riemann-Roch.- 3: Pseudo-holomorphic curves and applications.- V Some properties of holomorphic curves in almost complex manifolds.- VI Singularities and positivity of intersections of J-holomorphic curves.- VII Gromov's Schwarz lemma as an estimate of the gradient for holomorphic curves.- VIII Compactness.- IX Exemples de courbes pseudo-holomorphes en géométrie riemannienne.- X Symplectic rigidity: Lagrangian submanifolds.- Authors' addresses.