Unlike in the Riemannian case, symplectic manifolds have no local invariants such as curvature. Another difference with Riemannian geometry is that not every differentiable manifold need admit a symplectic form; there are certain topological restrictions. For example, every symplectic manifold is even-dimensional and orientable. Additionally, if M is a closed symplectic manifold, then the 2nd de Rham cohomology group H2 M is nontrivial; this implies, for example, that the only n-sphere that admits a symplectic form is the 2-sphere. A parallel that one can draw between the two subjects is the analogy between geodesics in Riemannian geometry and pseudoholomorphic curves in symplectic geometry: Geodesics are curves of shortest length locally , while pseudoholomorphic curves are surfaces of minimal area.
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Oxford Graduate Texts in Mathematics Can be used as an introductory textbook for graduate students or mature mathematicians who are interested in symplectic topology but are as yet unfamiliar with the subject. All chapters have been revised to improve the exposition and to make them more readable, new material has been added in many places, and various proofs have been tightened up.
Copious new references to key papers have been added to the bibliography. Chapter 13 has been completely rewritten and has a new title Questions of Existence and Uniqueness. A new Chapter 14 on open problems in the field has been added.
New to this Edition: All chapters have been revised to improve the exposition and to make them more readable, new material has been added in many places, and various proofs have been tightened up. A section on GIT has been added to Chapter 5. The material on contact geometry in Chapter 3 has been expanded. It now contains an introduction to existence and uniqueness problems in symplectic topology, a section an various examples, an overview of Taubes-Seiberg-Witten theory and its applications to symplectic topology, and a section on symplectic 4-manifolds.
Introduction to Symplectic Topology.
Introduction to Symplectic Topology
Dusa McDuff and Dietmar Salamon Abstract Over the past number of years powerful new methods in analysis and topology have led to the development of the modern global theory of symplectic topology, including several striking and important results. The first edition of Introduction to Symplectic Topology was published in The book was the first comprehensive introduction to the subject and became a key text in the area. In , a significantly revised second edition contained new sections and updates. This third edition includes both further updates and new material on this fast-developing area. All chapters have been revised to
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