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 1Iaslov 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.
1 Introduction.- 2 Darboux' Theorem and Examples of Symplectic Manifolds.- 3 Generating Functions.- 4 Symplectic Capacities.- 5 Floer Homology.- 6 Pseudoholomorphic Curves.- 7 Gromov's Compactness Theorem from a Geometrical Point of View.- 8 Contact structures.- A Generalities on Homology and Cohomology.- A.1 Axioms for homology.- A.2 Axioms for cohomology.- A.3 Homomorphisms of (co)homology sequences.- A.4 The (co)homology sequence of a triple.- A.5 Homotopy equivalence and contractibility.- A.6 Direct sums.- A.7 Triads.- A.8 Mayer-Vietoris sequence of a triad.- References.
