symplectic manifold

symplectic manifold
Name	symplectic manifold
Dimension	2n
Structure	symplectic form

symplectic manifold A symplectic manifold is a smooth even-dimensional manifold equipped with a closed nondegenerate 2-form, central to modern Isaac Newton-inspired Classical mechanics formulations and influencing research by Henri Poincaré, Élie Cartan, Andrey Kolmogorov, Vladimir Arnold, and Mikhail Gromov. It provides the geometric language for the Hamiltonian mechanics used in problems studied at institutions such as École Normale Supérieure, Princeton University, University of Cambridge, and Massachusetts Institute of Technology. Research developments involving symplectic manifolds appear in work connected to prizes like the Fields Medal awarded to researchers influencing Arnold-style problems and emerging in seminars at Institute for Advanced Study.

Definition and basic properties

A compact oriented 2n-dimensional smooth manifold M together with a closed nondegenerate 2-form ω is the basic object studied by mathematicians including William Thurston, Jean-Pierre Serre, John Milnor, and Michael Atiyah. The nondegeneracy of ω means that the interior product with ω gives an isomorphism between the tangent bundle and the cotangent bundle, a property invoked in expositions by Raoul Bott, Isadore Singer, Shing-Tung Yau, and Simon Donaldson. Closedness (dω = 0) relates to cohomology classes in the sense developed by Henri Cartan and used in calculations popularized by groups at Courant Institute and Harvard University. Orientation induced by ω plays a role in intersection-theoretic frameworks studied by Alexander Grothendieck and Jean-Pierre Kahane.

Examples

Standard examples include Euclidean space R^{2n} with the canonical form studied in classical treatments connected to Gottfried Leibniz-inspired analysis and in lectures at University of Göttingen and University of Bonn. Cotangent bundles T*X of smooth manifolds X appear in applications taught at University of Chicago and in monographs by Vladimir Arnold and Jerrold Marsden. Complex projective spaces CP^n with the Fubini-Study form are central examples discussed in seminars at École Polytechnique and in work by Federigo Enriques-era scholars. Symplectic blow-ups and Gompf constructions are constructions developed by researchers linked to University of California, Berkeley and University of Warwick. Other examples arise from Kähler manifolds investigated by Calabi and Shing-Tung Yau and from moduli spaces appearing in studies by Pierre Deligne and Alexander Grothendieck.

Local structure and Darboux's theorem

Darboux's theorem, proven in contexts influenced by lectures at Université de Paris and historic expositions by Jean-Gaston Darboux, asserts that there exist local coordinates in which ω takes the standard form; this theorem underlies analyses by Henri Poincaré and later expositors at University of Oxford and Princeton University. The local triviality contrasts with global invariants studied by Mikhail Gromov and Glen Bredon and is exploited in normal form results appearing in courses at Massachusetts Institute of Technology and ETH Zurich. Darboux charts are foundational in constructions used by Michel Herman and techniques developed in workshops at Max Planck Institute for Mathematics.

Symplectic topology and invariants

Global invariants such as Gromov–Witten invariants were introduced by Mikhail Gromov and further developed by researchers associated with Institut des Hautes Études Scientifiques and Stanford University. Floer homology, initiated by Andreas Floer and extended by collaborators at University of Basel and ETH Zurich, yields powerful tools for distinguishing symplectic manifolds. Symplectic rigidity phenomena, including Gromov's non-squeezing theorem, were milestones presented in venues like International Congress of Mathematicians and elaborated by scholars at University of Chicago and Imperial College London. Quantum cohomology, developed by teams at California Institute of Technology and University of Michigan, links to enumerative geometry traditions stemming from David Mumford and William Fulton.

Relations to complex and Kähler geometry

When a symplectic form ω is compatible with an integrable almost complex structure J, the manifold becomes Kähler, a context explored in foundational work by Élie Cartan, Shing-Tung Yau, and Aubin. Kähler manifolds like complex tori and Calabi–Yau varieties play roles in research at Institute for Advanced Study and Harvard University, connecting symplectic techniques to results by Maxwell, Hermann Weyl, and André Weil. The interplay of Hodge theory from Jean-Pierre Serre and mirror symmetry developments involving Cumrun Vafa and Edward Witten highlight cross-disciplinary links present in conferences at CERN and Simons Center for Geometry and Physics.

Applications in physics and Hamiltonian dynamics

Symplectic manifolds model phase spaces in Hamiltonian dynamics studied by Isaac Newton-influenced traditions and modern expositions by Vladimir Arnold and Jerrold Marsden. Canonical transformations correspond to symplectomorphisms discussed in lectures at Princeton University and Stanford University. In mathematical formulations of classical field theories and aspects of quantum field theory, symplectic techniques appear in works by Edward Witten and in research programs at Perimeter Institute and CERN. The use of symplectic reduction, as in the Marsden–Weinstein theorem, connects to mechanics problems addressed by teams at California Institute of Technology and University of Cambridge.

Category:Symplectic geometry