Symplectic Geometry

Abstract

There are two essential differences between symplectic and Riemannian geometries. First, the Riemannian geometry is “rigid” in the sense that two Riemannian manifolds chosen at random are most likely to be locally nonisometric. On the contrary, any two symplectic manifolds are locally isometric in the sense that the symplectic 2-form on any symplectic manifold always takes the canonical form of Example 1.1.2 in appropriate local coordinates, due to Darboux’s theorem [GS1].