Symplectic manifold


In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold,, equipped with a closed nondegenerate differential 2-form, called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology. Symplectic manifolds arise naturally in abstract formulations of classical mechanics and analytical mechanics as the cotangent bundles of manifolds. For example, in the Hamiltonian formulation of classical mechanics, which provides one of the major motivations for the field, the set of all possible configurations of a system is modeled as a manifold, and this manifold's cotangent bundle describes the phase space of the system.

Motivation

Symplectic manifolds arise from classical mechanics; in particular, they are a generalization of the phase space of a closed system. In the same way the Hamilton equations allow one to derive the time evolution of a system from a set of differential equations, the symplectic form should allow one to obtain a vector field describing the flow of the system from the differential dH of a Hamiltonian function H. So we require a linear map, or equivalently, an element of. Letting ω denote a section of, the requirement that ω be non-degenerate ensures that for every differential dH there is a unique corresponding vector field VH such that. Since one desires the Hamiltonian to be constant along flow lines, one should have, which implies that ω is alternating and hence a 2-form. Finally, one makes the requirement that ω should not change under flow lines, i.e. that the Lie derivative of ω along VH vanishes. Applying Cartan's formula, this amounts to :
so that, on repeating this argument for different smooth functions such that the corresponding span the tangent space at each point the argument is applied at, we see that the requirement for the vanishing Lie derivative along flows of corresponding to arbitrary smooth is equivalent to the requirement that ω should be closed.

Definition

A symplectic form on a smooth manifold is a closed non-degenerate differential 2-form. Here, non-degenerate means that for every point, the skew-symmetric pairing on the tangent space defined by is non-degenerate. That is to say, if there exists an such that for all, then. Since in odd dimensions, skew-symmetric matrices are always singular, the requirement that be nondegenerate implies that has an even dimension. The closed condition means that the exterior derivative of vanishes. A symplectic manifold is a pair where is a smooth manifold and is a symplectic form. Assigning a symplectic form to is referred to as giving a symplectic structure.

Linear symplectic manifold

There is a standard linear model, namely a symplectic vector space Let be a basis for We define our symplectic form ω on this basis as follows:
In this case the symplectic form reduces to a simple quadratic form. If In denotes the n × n identity matrix then the matrix, Ω, of this quadratic form is given by the block matrix:

Lagrangian and other submanifolds

There are several natural geometric notions of submanifold of a symplectic manifold.
The most important case of the isotropic submanifolds is that of Lagrangian submanifolds. A Lagrangian submanifold is, by definition, an isotropic submanifold of maximal dimension, namely half the dimension of the ambient symplectic manifold. One major example is that the graph of a symplectomorphism in the product symplectic manifold is Lagrangian. Their intersections display rigidity properties not possessed by smooth manifolds; the Arnold conjecture gives the sum of the submanifold's Betti numbers as a lower bound for the number of self intersections of a smooth Lagrangian submanifold, rather than the Euler characteristic in the smooth case.

Examples

Let have global coordinates labelled Then, we can equip with the canonical symplectic form
There is a standard Lagrangian submanifold given by. The form vanishes on because given any pair of tangent vectors we have that To elucidate, consider the case. Then, and Notice that when we expand this out
both terms we have a factor, which is 0, by definition.
The cotangent bundle of a manifold is locally modeled on a space similar to the first example. It can be shown that we can glue these affine symplectic forms hence this bundle forms a symplectic manifold. A more non-trivial example of a Lagrangian submanifold is the zero section of the cotangent bundle of a manifold. For example, let
Then, we can present as
where we are treating the symbols as coordinates of We can consider the subset where the coordinates and, giving us the zero section. This example can be repeated for any manifold defined by the vanishing locus of smooth functions and their differentials.
Another useful class of Lagrangian submanifolds can be found using Morse theory. Given a Morse function and for a small enough one can construct a Lagrangian submanifold given by the vanishing locus. For a generic morse function we have a Lagrangian intersection given by.

Special Lagrangian submanifolds

In the case of Kahler manifolds we can make a choice on as a holomorphic n-form, where is the real part and imaginary. A Lagrangian submanifold is called special if in addition to the above Lagrangian condition the restriction to is vanishing. In other words, the real part restricted on leads the volume form on. The following examples are known as special Lagrangian submanifolds,
The SYZ conjecture has been proved for special Lagrangian submanifolds but in general, it is open, and brings a lot of impacts to the study of mirror symmetry. see

Lagrangian fibration

A Lagrangian fibration of a symplectic manifold M is a fibration where all of the fibres are Lagrangian submanifolds. Since M is even-dimensional we can take local coordinates and by Darboux's theorem the symplectic form ω can be, at least locally, written as, where d denotes the exterior derivative and ∧ denotes the exterior product. Using this set-up we can locally think of M as being the cotangent bundle and the Lagrangian fibration as the trivial fibration This is the canonical picture.

Lagrangian mapping

Let L be a Lagrangian submanifold of a symplectic manifold given by an immersion . Let give a Lagrangian fibration of K. The composite is a Lagrangian mapping. The critical value set of π ∘ i is called a caustic.
Two Lagrangian maps and are called Lagrangian equivalent if there exist diffeomorphisms σ, τ and ν such that both sides of the diagram given on the right commute, and τ preserves the symplectic form. Symbolically:
where τ∗ω2 denotes the pull back of ω2 by τ.

Special cases and generalizations