The Symplectic manifold reference article from the English Wikipedia on 24-Apr-2004 (provided by Fixed Reference: snapshots of Wikipedia from wikipedia.org)

Symplectic manifold

Symplectic topology is that part of mathematics concerned with the study of symplectic manifolds. These manifolds arise naturally in the Hamiltonian formulation of classical mechanics, which provides one of the major motivations for the field. There is a standard 'local' model, namely R²ⁿ with and for all i,j=1,...,n-1 where δ is the Kronecker delta function. This is called is a linear symplectic space.

A symplectic manifold is a pair (M,ω) where M is a smooth manifold and ω is a closed, nondegenerate, 2-form on M called the symplectic form. Here, "nondegenerate" means that for every nonzero vector u in the tangent space at a point, there is a vector v such that

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.

Table of contents

1 Hamiltonian vector fields 2 Symplectomorphisms 3 Pseudoholomorphic curves 4 Related topics 5 References

Hamiltonian vector fields

On a symplectic manifold, every differentiable function, H, defines a unique vector field, X_H, called a Hamiltonian vector field. 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)

(Warning: other sign conventions are in use).

Symplectomorphisms

The flow of a Hamiltonian vector field 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 vector fields. 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 -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.

Pseudoholomorphic curves

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; this result has led to the development of a fairly large subdiscipline of symplectic topology. Results arising from Gromov's theory include Gromov's nonsqueezing theorem concerning symplectic embeddings of spheres into cylinders, as well as a conjecture of Vladimir Arnol'd concerning the number of fixed points of Hamiltonian flows — this was 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.

Related topics

• Hamiltonian mechanics
• Riemannian geometry
• contact geometry
• symplectic group, symplectic matrix

References

• McDuff, D. and Salamon, D.: Introduction to Symplectic Topology (Oxford Mathematical Monographs)