Symplectic manifold

Symplectic topology is that part of mathematics concerned with the study of symplectic manifolds (also called symplectic space). These manifolds arise in the Hamiltonian formulation of classical mechanics, which provides one of the major motivations for the field. Symplectic spaces are also related to contact geometry.

A symplectic manifold is a pair (M,ω) of a smooth manifold M together with a closed non-degenerate differential 2-form ω, the symplectic form. "Non-degenerate" means that for every vector u in the tangent space at a point, there is a vector v such that the skew product

ω(u,v) ≠ 0

Fundamental examples of symplectic manifolds are given by the cotangent bundles of manifolds; these arise in classical mechanics, where the set of all possible configurations of a system is modelled as a manifold, and this manifold's cotangent bundle describes the phase space of the system. Kähler manifolds are also symplectic manifolds. Well into the 1970's, symplectic experts were unsure of whether any compact non-Kähler symplectic manifolds existed, but since then many examples have been constructed; in particular, Robert Gompf has shown that every finitely presented group occurs as the fundamental group of some symplectic 4-manifold, in marked contrast with the Kähler case.

Directly from the definition, one can show that M is of even dimension 2n and that ωⁿ is a nowhere vanishing form, the symplectic volume form. It follows that a symplectic manifold is canonically oriented and comes with a canonical measure, the Liouville measure.

On a symplectic manifold, every differentiable function, H, defines a unique Hamiltonian vector field X_H. It is defined such that for every vector field Y on M the identity

dH(Y) = ω(X_H,Y)

holds. The Hamiltonian vector fields give the functions on M the structure of a Lie algebra with bracket the Poisson bracket

{f,g} = ω(X_f,X_g) = X_g(f)

(other sign conventions are also in use)

The flow of a Hamiltonian vectorfield is a symplectomorphism i.e. a diffeomorphism that preserves the symplectic form. This follows from the closedness of the symplectic form and the expression of the Lie derivative in terms of the exterior derivative. As a direct consequence we have Liouville's theorem: the symplectic volume is invariant under a Hamiltionan flow. Since {H,H} = X_H(H) = 0 the flow of a Hamiltonian vector field also preserves H. In physics this is interpreted as the law of conservation of energy. Liouville's theorem is interpreted as the conservation of phase volume in Hamiltonian systems, which is the basis for classical statistical mechanics. We just showed that there is a one-to-one correspondence between infinitesimal symplectomorphisms and smooth functions over a symplectic manifold.

Unlike Riemannian manifolds, symplectic manifolds are extremely non-rigid: they have many symplectomorphisms coming from Hamiltonian vectorfields. The fundamental difference between Riemannian and Symplectic geometry is that a symplectic manifold has no local invariants: according to Darboux's theorem for every point x in a symplectic manifold there is a local coordinate system called action-angle with coordinates p₁,...,p_n, q₁,...,q_n, such that

ω = ∑ dp_i ∧ dq_i

Finite-dimensional subgroups of the group of symplectomorphisms are Lie groups. Representations of these Lie groups (after ℏ-deformations in general!) on Hilbert spaces are called "quantizations". When the Lie group is the one defined by a Hamiltonian, it is called a "quantization by energy". The corresponding Lie operator from the Lie algebra to the Lie algebra of continuous linear operators is also sometimes called the quantization, and is a more common way of looking at it among physicists.

Although most symplectic manifolds are not Kähler and so do not have an integrable complex structure compatible with the symplectic form, Mikhail Gromov 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 complex manifold except the requirement that the transition functions be holomorphic. A Riemann surface mapped into a symplectic manifold compatibly with the almost complex structure is called a pseudoholomorphic curve, and Gromov proved a compactness theorem for such curves which has led to the development of a fairly large subdiscipline of symplectic topology. Results leading from Gromov's theory include Gromov's nonsqueezing theorem concerning symplectic embeddings of spheres into cylinders, as well as a conjecture of Vladimir Arnold concerning the number of fixed points of Hamiltonian flows, proven in increasing generality by several researchers beginning with Andreas Floer, who introduced what is now known as Floer homology using Gromov's methods. Pseudoholomorphic curves are also a source of symplectic invariants, known as Gromov-Witten invariants, by which two different symplectic manifolds could in principle be distinguished.

The relation between symplectic geometry and Hamiltonian mechanics should explained in more detail.

Symplectic capcities should be mentioned.

Examples would be nice. Why are these things studied? I suspect because of physics?

References

Mc Duff D., Salomon D.: Introduction to Symplectic Topology (Oxford Mathematical Monographs)