Symplectic Structures— A New Approach to Geometry

Introduction Symplectic geometry is the geometry of a closed skew-symmetric form. It turns out to be very different from the Riemannian geometry with which we are familiar. One important difference is that, although all its concepts are initially expressed in the smooth category (for example, in terms of differential forms), in some intrinsic way they do not involve derivatives. Thus symplectic geometry is essentially topological in nature. Indeed, one often talks about symplectic topology. Another important feature is that it is a 2-dimensional geometry that measures the area of complex curves instead of the length of real curves. The classical geometry over the complex numbers is Kähler geometry, the geometry of a complex manifold with a compatible Riemannian metric. This is a very rich geometry with a detailed local structure. In contrast, symplectic geometry is flabby, though as should become clear, not completely flabby—there are interesting elements of global structure. The comparison can be roughly stated as follows: { Kähler rich detail }