Symplectic Manifold - Lagrangian and Other Submanifolds

Lagrangian and Other Submanifolds

There are several natural geometric notions of submanifold of a symplectic manifold.

• symplectic submanifolds (potentially of any even dimension) are submanifolds where the symplectic form is required to induce a symplectic form on them.
• isotropic submanifolds are submanifolds where the symplectic form restricts to zero, i.e. each tangent space is an isotropic subspace of the ambient manifold's tangent space. Similarly, if each tangent subspace to a submanifold is co-isotropic (the dual of an isotropic subspace), the submanifold is called co-isotropic.

The most important case of the isotropic submanifolds is that of Lagrangian submanifolds. A Lagrangian submanifold is, by definition, an isotropic submanifold of maximal dimension, namely half the dimension of the ambient symplectic manifold. Lagrangian submanifolds arise naturally in many physical and geometric situations. One major example is that the graph of a symplectomorphism in the product symplectic manifold (M × M, ω × −ω) is Lagrangian. Their intersections display rigidity properties not possessed by smooth manifolds; the Arnold conjecture gives the sum of the submanifold's Betti numbers as a lower bound for the number of self intersections of a smooth Lagrangian submanifold, rather than the Euler characteristic in the smooth case.

Lagrangian submanifolds arise naturally in many physical and geometric situations. We shall see below that caustics can be explained in terms of Lagrangian submanifolds.