Synopsis:
The seminar Symplectic Geometry at the University of Berne in summer 1992 showed that the topic of this book is a very active field, where many different branches of mathematics come tog9ther: differential geometry, topology, partial differential equations, variational calculus, and complex analysis. As usual in such a situation, it may be tedious to collect all the necessary ingredients. The present book is intended to give the nonspecialist a solid introduction to the recent developments in symplectic and contact geometry. Chapter 1 gives a review of the symplectic group Sp(n,R), sympkctic manifolds, and Hamiltonian systems (last but not least to fix the notations). The 1\Iaslov index for closed curves as well as arcs in Sp(n, R) is discussed. This index will be used in chapters 5 and 8. Chapter 2 contains a more detailed account of symplectic manifolds start ing with a proof of the Darboux theorem saying that there are no local in variants in symplectic geometry. The most important examples of symplectic manifolds will be introduced: cotangent spaces and Kahler manifolds. Finally we discuss the theory of coadjoint orbits and the Kostant-Souriau theorem, which are concerned with the question of which homogeneous spaces carry a symplectic structure.
Synopsis:
With the publication of Gromov's seminal paper in 1985 and with the development of Floer homology in the eighties, symplectic geometry has changed drastically. The complexity and diversity of the topics enriched this theory, but at the same time posed obstacles to easy comprehension. The present volume collects the basic building blocks of the theory and tries to present them in a readable form. The exposition is addressed towards advanced students, graduate students, mathematicians and physicists and all those who want to gain an insight into this fascinating and rapidly developing field. The present volume is intended as a first reading in symplectic geometry. Basic notions are explained and the principal examples are outlined. Capacities, generating functions, Floer homology, pseudoholomorphic curves and contact structures make up the core of the exposition. To a large extent the treatment is self-contained, and references to the current literature facilitate the access to the original papers.
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