Symplectic geometry
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Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2form. Symplectic geometry has its origins in the Hamiltonian formulation of classical mechanics where the phase space of certain classical systems takes on the structure of a symplectic manifold.
Introduction
A symplectic geometry is defined on a smooth evendimensional space that is a differentiable manifold. On this space is defined a geometric object, the symplectic form, that allows for the measurement of sizes of twodimensional objects in the space. The symplectic form in symplectic geometry plays a role analogous to that of the metric tensor in Riemannian geometry. Where the metric tensor measures lengths and angles, the symplectic form measures areas.^{}
Symplectic geometry arose from the study of classical mechanics and an example of a symplectic structure is the motion of an object in one dimension. To specify the trajectory of the object, one requires both the position q and the momentum p, which form a point (p,q) in the Euclidean plane ℝ^{2}. In this case, the symplectic form is
and is an area form that measures the area A of a region S in the plane through integration:
The area is important because as conservative dynamical systems evolve in time, this area is invariant.^{}
Higher dimensional symplectic geometries are defined analogously. A 2ndimensional symplectic geometry is formed of pairs of directions
in a 2ndimensional manifold along with a symplectic form
This symplectic form yields the size of a 2ndimensional region V in the space as the sum of the areas of the projections of V onto each of the planes formed by the pairs of directions^{}
Comparison with Riemannian geometry
Symplectic geometry has a number of similarities with and differences from Riemannian geometry, which is the study of differentiable manifolds equipped with nondegenerate, symmetric 2tensors (called metric tensors). Unlike in the Riemannian case, symplectic manifolds have no local invariants such as curvature. This is a consequence of Darboux's theorem which states that a neighborhood of any point of a 2ndimensional symplectic manifold is isomorphic to the standard symplectic structure on an open set of ℝ^{2n}. 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 evendimensional and orientable. Additionally, if M is a closed symplectic manifold, then the 2nd de Rham cohomology group H^{2}(M) is nontrivial; this implies, for example, that the only nsphere that admits a symplectic form is the 2sphere.
Examples and structures
Every Kähler manifold is also a symplectic manifold. Well into the 1970s, symplectic experts were unsure whether any compact nonKähler symplectic manifolds existed, but since then many examples have been constructed (the first was due to William Thurston); in particular, Robert Gompf has shown that every finitely presented group occurs as the fundamental group of some symplectic 4manifold, in marked contrast with the Kähler case.
Most symplectic manifolds, one can say, are not Kähler; and so do not have an integrable complex structure compatible with the symplectic form. Mikhail Gromov, however, made the important observation that symplectic manifolds do admit an abundance of compatible almost complex structures, so that they satisfy all the axioms for a Kähler manifold except the requirement that the transition maps be holomorphic.
Gromov used the existence of almost complex structures on symplectic manifolds to develop a theory of pseudoholomorphic curves, which has led to a number of advancements in symplectic topology, including a class of symplectic invariants now known as Gromov–Witten invariants. These invariants also play a key role in string theory.
Name
The name "complex group" formerly advocated by me in allusion to line complexes, as these are defined by the vanishing of antisymmetric bilinear forms, has become more and more embarrassing through collision with the word "complex" in the connotation of complex number. I therefore propose to replace it by the corresponding Greek adjective "symplectic." Dickson calls the group the "Abelian linear group" in homage to Abel who first studied it.
Symplectic geometry is also called symplectic topology although the latter is really a subfield concerned with important global questions in symplectic geometry.
The term "symplectic" is a calque of "complex", introduced by Weyl (1939, footnote, p.165); previously, the "symplectic group" had been called the "line complex group". Complex comes from the Latin complexus, meaning "braided together" (co + plexus), while symplectic comes from the corresponding Greek symplektikos (συμπλεκτικός); in both cases the suffix comes from the IndoEuropean root *plek.^{} This naming reflects the deep connections between complex and symplectic structures.