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109 pages, 5 Mb, September 2004

Abstract:

Authors' Introduction quote: "This is an overview of symplectic geometry1 – the geometry of symplectic manifolds. From a language for classical mechanics in the XVIII century, symplectic geometry has matured since the 1960’s to a rich and central branch of differential geometry and topology. A current survey can thus only aspire to give a partial flavor on this exciting field. The following six topics have been chosen for this handbook: 1. Symplectic manifolds are manifolds equipped with symplectic forms. A symplectic form is a closed nondegenerate 2-form. The algebraic condition (nondegeneracy) says that the top exterior power of a symplectic form is a volume form, therefore symplectic manifolds are necessarily even-dimensional and orientable. The analytical condition (closedness) is a natural differential equation that forces all symplectic manifolds to being locally indistinguishable: they all locally look like an even-dimensional euclidean space equipped with the P dxi ^ dyi symplectic form. All cotangent bundles admit canonical symplectic forms, a fact relevant for analysis of differential operators, dynamical systems, classical mechanics, etc. Basic properties, major classical examples, equivalence notions, local normal forms of symplectic manifolds and symplectic submanifolds are discussed in Chapter 1. 2. Lagrangian submanifolds2 are submanifolds of symplectic manifolds of half dimension and where the restriction of the symplectic form vanishes identically. By the lagrangian creed [137], everything is a lagrangian submanifold, starting with closed 1-forms, real functions modulo constants and symplectomorphisms (diffeomorphisms that respect the symplectic forms). Chapter 2 also describes normal neighborhoods of lagrangian submanifolds with applications. 3. Complex structures or almost complex structures abound in symplectic geometry: any symplectic manifold possesses almost complex structures, and even so in a compatible sense. This is the point of departure for the modern technique of studying pseudoholomorphic curves, as first proposed by Gromov [64]. K¨ahler geometry lies at the intersection of complex, riemannian and symplectic geometries, and plays a central role in these three fields. Chapter 3 includes the local normal form for K¨ahler manifolds and a summary of Hodge theory for K¨ahler manifolds. 4. Symplectic geography is concerned with existence and uniqueness of symplectic forms on a given manifold. Important results from K¨ahler geometry remain true in the more general symplectic category, as shown using pseudoholomorphic methods. This viewpoint was more recently continued with work on the existence of certain symplectic submanifolds, in the context of Seiberg-Witten invariants, and with topological descriptions in terms of Lefschetz pencils. Both of these directions 1The word symplectic in mathematics was coined in the late 1930’s by Weyl [142, p.165] who substituted the Latin root in complex by the corresponding Greek root in order to label the symplectic group (first studied be Abel). An English dictionary is likely to list symplectic as the name for a bone in a fish’s head. 2The name lagrangian manifold was introduced by Maslov [93] in the 1960’s, followed by lagrangian plane, etc., introduced by Arnold [2]."

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Open access: http://www.math.princeton.edu/~acannas/symplectic.pdf

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