"Over the past 15 years symplectic geometry has developed its own identity, and can now stand alongside traditional Riemannian geometry as a rich and meaningful part of mathematics. The basic definitions are very natural from a mathematical point of view: one studies the geometry of a skew-symmetric bilinear form ω rather than a symmetric one. However, this seemingly innocent change of symmetry has radical effects. For example, one dimensional measurements vanish since ω(v, v) = −ω(v, v) by skew-symmetry. ... The theory has two faces. There are two kinds of geometric subobjects in a symplectic manifolds, hypersurfaces and Lagrangian submanifolds that appear in dynamical constructions, and even-dimensional symplectic submanifolds that are closely related to Riemannian and complex geometry. As we shall see, the analog of a geodesic in a symplectic manifold is a two-dimensional surface called a ."