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AuthorSilva, Ana Cannas da. author
TitleLectures on Symplectic Geometry [electronic resource] / by Ana Cannas da Silva
ImprintBerlin, Heidelberg : Springer Berlin Heidelberg, 2001
Connect tohttp://dx.doi.org/10.1007/978-3-540-45330-7
Descript XII, 220 p. online resource

SUMMARY

The goal of these notes is to provide a fast introduction to symplectic geometry for graduate students with some knowledge of differential geometry, de Rham theory and classical Lie groups. This text addresses symplectomorphisms, local forms, contact manifolds, compatible almost complex structures, Kaehler manifolds, hamiltonian mechanics, moment maps, symplectic reduction and symplectic toric manifolds. It contains guided problems, called homework, designed to complement the exposition or extend the reader's understanding. There are by now excellent references on symplectic geometry, a subset of which is in the bibliography of this book. However, the most efficient introduction to a subject is often a short elementary treatment, and these notes attempt to serve that purpose. This text provides a taste of areas of current research and will prepare the reader to explore recent papers and extensive books on symplectic geometry where the pace is much faster. For this reprint numerous corrections and clarifications have been made, and the layout has been improved


CONTENT

Symplectic Manifolds -- Symplectic Forms -- Symplectic Form on the Cotangent Bundle -- Symplectomorphisms -- Lagrangian Submanifolds -- Generating Functions -- Recurrence -- Local Forms -- Preparation for the Local Theory -- Moser Theorems -- Darboux-Moser-Weinstein Theory -- Weinstein Tubular Neighborhood Theorem -- Contact Manifolds -- Contact Forms -- Contact Dynamics -- Compatible Almost Complex Structures -- Almost Complex Structures -- Compatible Triples -- Dolbeault Theory -- Kรคhler Manifolds -- Complex Manifolds -- Kรคhler Forms -- Compact Kรคhler Manifolds -- Hamiltonian Mechanics -- Hamiltonian Vector Fields -- Variational Principles -- Legendre Transform -- Moment Maps -- Actions -- Hamiltonian Actions -- Symplectic Reduction -- The Marsden-Weinstein-Meyer Theorem -- Reduction -- Moment Maps Revisited -- Moment Map in Gauge Theory -- Existence and Uniqueness of Moment Maps -- Convexity -- Symplectic Toric Manifolds -- Classification of Symplectic Toric Manifolds -- Delzant Construction -- Duistermaat-Heckman Theorems


Mathematics Partial differential equations Differential geometry Mathematics Differential Geometry Partial Differential Equations



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