Generating Functions, Symplectic Geometry, and Applications

A symplectic form on a manifold is a closed two form ω, nondegenerate as a skew-symmetric bilinear form on the tangent space at each point. Integration of the form on a two-dimensional submanifold S with boundary ∂S in M associates to S a real number (positive or negative) the “area of S”, which due to Stoke’s formula only depends on the curves ∂S, and the homology class of S rel ∂S. If moreover the form is exact, that is ω = dλ, the area of S is obtained by integrating λ over ∂S. In this case it is also possible to integrate λ on loops nonhomologous to zero and we get the notion of “area enclosed by a loop”. However this area depends on the choice of λ. If this choice is fixed once for all, we shall then talk about an exact manifold. One should be careful about the fact that this notion is slightly different from that of a symplectic manifold with exact symplectic form (because in the latter case we have not chosen the primitive of ω).