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.
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.