Symplectic manifold

In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, formula_1, equipped with a closed nondegenerate differential 2-form formula_2, 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 (here formula_3 is the interior product): 
so that, on repeating this argument for different smooth functions formula_5 such that the corresponding formula_6 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 formula_6 corresponding to arbitrary smooth formula_5 is equivalent to the requirement that "ω" should be closed.
Definition.
A symplectic form on a smooth manifold formula_1 is a closed non-degenerate differential 2-form formula_2. Here, non-degenerate means that for every point formula_11, the skew-symmetric pairing on the tangent space formula_12 defined by formula_2 is non-degenerate. That is to say, if there exists an formula_14 such that formula_15 for all formula_16, then formula_17. Since in odd dimensions, skew-symmetric matrices are always singular, the requirement that formula_2 be nondegenerate implies that formula_1 has an even dimension. The closed condition means that the exterior derivative of formula_2 vanishes. A symplectic manifold is a pair formula_21 where formula_1 is a smooth manifold and formula_2 is a symplectic form. Assigning a symplectic form to formula_1 is referred to as giving formula_1 a symplectic structure.
Examples.
Symplectic vector spaces.
Let formula_26 be a basis for formula_27 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:
Cotangent bundles.
Let formula_30 be a smooth manifold of dimension formula_31. Then the total space of the cotangent bundle formula_32 has a natural symplectic form, called the Poincaré two-form or the canonical symplectic form
Here formula_34 are any local coordinates on formula_30 and formula_36 are fibrewise coordinates with respect to the cotangent vectors formula_37. Cotangent bundles are the natural phase spaces of classical mechanics. The point of distinguishing upper and lower indexes is driven by the case of the manifold having a metric tensor, as is the case for Riemannian manifolds. Upper and lower indexes transform contra and covariantly under a change of coordinate frames. The phrase "fibrewise coordinates with respect to the cotangent vectors" is meant to convey that the momenta formula_38 are "soldered" to the velocities formula_39. The soldering is an expression of the idea that velocity and momentum are colinear, in that both move in the same direction, and differ by a scale factor.
Kähler manifolds.
A Kähler manifold is a symplectic manifold equipped with a compatible integrable complex structure. They form a particular class of complex manifolds. A large class of examples come from complex algebraic geometry. Any smooth complex projective variety formula_40 has a symplectic form which is the restriction of the Fubini-Study form on the projective space formula_41.
Lagrangian and other submanifolds.
There are several natural geometric notions of submanifold of a symplectic manifold formula_21.
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 formula_52 have global coordinates labelled formula_53 Then, we can equip formula_54 with the canonical symplectic form 
There is a standard Lagrangian submanifold given by formula_56. The form formula_48 vanishes on formula_58 because given any pair of tangent vectors formula_59 we have that formula_60 To elucidate, consider the case formula_61. Then, formula_62 and formula_63 Notice that when we expand this out
both terms we have a formula_65 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 formula_67 as
where we are treating the symbols formula_69 as coordinates of formula_70 We can consider the subset where the coordinates formula_71 and formula_72, giving us the zero section. This example can be repeated for any manifold defined by the vanishing locus of smooth functions formula_73 and their differentials formula_74.
Another useful class of Lagrangian submanifolds can be found using Morse theory. Given a Morse function formula_75 and for a small enough formula_76 one can construct a Lagrangian submanifold given by the vanishing locus formula_77. For a generic morse function we have a Lagrangian intersection given by formula_78.
Special Lagrangian submanifolds.
In the case of Kahler manifolds (or Calabi-Yau manifolds) we can make a choice formula_79 on formula_80 as a holomorphic n-form, where formula_81 is the real part and formula_82 imaginary. A Lagrangian submanifold formula_83 is called special if in addition to the above Lagrangian condition the restriction formula_82 to formula_83 is vanishing. In other words, the real part formula_81 restricted on formula_83 leads the volume form on formula_83. 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. This form is called the Poincaré two-form or the canonical two-form. Using this set-up we can locally think of "M" as being the cotangent bundle formula_89 and the Lagrangian fibration as the trivial fibration formula_90 This is the canonical picture.
Lagrangian mapping.
Let "L" be a Lagrangian submanifold of a symplectic manifold ("K",ω) given by an immersion ("i" is called a Lagrangian 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"∗"ω"2 denotes the pull back of "ω"2 by "τ".