Symplectic Geometry

Held as a seminar at the University of Bern in the summer of 1992, the material behind this volume reflects how lively symplectic geometry has become, drawing together differential geometry, topology, partial differential equations, variational calculus and complex analysis. Because the field spans so many areas, newcomers often struggle to gather the background they need. This book aims to offer non-specialists a sound entry point into recent progress in symplectic and contact geometry.

The first chapter reviews the symplectic group Sp(n,R), symplectic manifolds and Hamiltonian systems, and also settles the notation used throughout. It then treats the Maslov index for closed curves and for arcs in Sp(n,R), a tool that returns in later chapters. The second chapter examines symplectic manifolds in more depth, beginning with a proof of the Darboux theorem, which states that symplectic geometry has no local invariants. Cotangent spaces and Kähler manifolds are presented as the central examples, and the chapter closes with coadjoint orbits and the Kostant-Souriau theorem, which determines which homogeneous spaces can carry a symplectic structure.