Let be a symplectic manifold of dim , a manifold smoothly embedded by a map is called a Lagrange submanifold of if the pull-back to of the symplectic form on by vanishes on
and is of maximal possible dimension compatible with the symplectic structure , i.e. dim .
Let be a contact manifold of dimension , a -dimensional manifold such that
with an embedding, is called a Legendre submanifold of .