The Symplectic Manifold

I take my information from The Road to Reality and Wikipedia’s Symplectic geometry and especially this version which includes the origin of the name “symplectic”. The Symplectization of Science too has been useful but see bug.

See some confusing related names.

The dot product of two differential vectors dxi and dyi is gijdxidyj where g is the metric tensor which is symmetric; gij = gji. By contrast we consider the scalar expression ωijdxidyj where ω is anti-symmetric, ωij + ωji = 0. Both gij and ωij are functions of the coördinates xi. This ω field endows the space with a sense of oriented area of a surface or 2D submanifold. An equivalent spec is an anti-symmetric bilinear differential form. Either of these assigns a scalar area to any smooth 2D submanifold.

A bilinear form Ω(x, y) is degenerate iff there is a non-zero vector x such that for all vectors y Ω(x, y) = 0. note
An anti-symmetric bilinear form Ω(x, y) is closed iff the area of the boundary of any 3D submanifold is 0. note

All anti-symmetric bilinear forms in an odd number of dimensions are degenerate.

A symplectic manifold is defined as a manifold with a non degenerate closed anti-symmetric bilinear form.

Darboux proved that for every point of a symplectic manifold, there is a neighborhood of that point which is isomorphic to the standard unit ball in the flat symplectic space of the same dimensions; there is no local character.

Penrose calls symplectic manifolds ‘floppy’ for this reason. By contrast, the Riemannian manifolds has a curvature tensor, computed from the metric, which is inherent, non-zero and thus distinguishes one point from another in such spaces. It is only asymptotically flat.

The following is very wrong but can be fixed!!
The vector dual of a 2D area is a ‘Grassmann field’ of the sort that be wrapped in a string. In 2D you can wrap a set of points in a string; in 3D you can wrap a stalk of celery (1D stuff) in a string; in nD you can wrap a stalk of (n−2)D things. Each of these stalks lack boundaries, just as Faraday’s magnet lines of force didn’t come to an end.

ω is non−degenerate iff ∀u(u∊V → (∀v(v∊V → ω(u, v) = 0) → u = 0))
ω is non−degenerate iff ∀u(u∊V & u ≠ 0 → ∃v(v∊V & ω(u, v) ≠ 0))
ω is degenerate iff ∃u(u∊V & u ≠ 0 → (∀v(v∊V → ω(u, v) = 0))
An explicit construction of Gromov’s ‘exotic’ sympletic structure.