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. Lagrangian submanifolds of a symplectic manifold are submanifolds where the restriction of the symplectic form to is vanishing, i.e. and . Lagrangian submanifolds are the maximal isotropic submanifolds. One major example is that the graph of a symplectomorphism in the product symplectic manifold 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. Examples Let have global coordinates labelled . Then, we can equip with the canonical symplectic form There is a standard Lagrangian submanifold given by . The form vanishes on because given any pair of tangent vectors we have that To elucidate, consider the case . Then, and . Notice that when we expand this out both terms we have a factor, which is 0, by definition. Example: Cotangent bundle The cotangent bundle of a manifold is locally modeled on a space similar to the first example. It can be shown that we can glue these affine symplectic forms hence this bundle forms a symplectic manifold. A less trivial example of a Lagrangian submanifold is the zero section of the cotangent bundle of a manifold. For example, let Then, we can present as where we are treating the symbols as coordinates of . We can consider the subset where the coordinates and , giving us the zero section. This example can be repeated for any manifold defined by the vanishing locus of smooth functions and their differentials . Example: Parametric submanifold Consider the canonical space with coordinates . A parametric submanifold of is one that is parameterized by coordinates such that This manifold is a Lagrangian submanifold if the Lagrange bracket vanishes for all . That is, it is Lagrangian if for all . This can be seen by expanding in the condition for a Lagrangian submanifold . This is that the symplectic form must vanish on the tangent manifold ; that is, it must vanish for all tangent vectors: for all . Simplify the result by making use of the canonical symplectic form on : and all others vanishing. As local charts