Legendrian submanifold

Summary

A Legendrian submanifold of a contact manifold of dimension $2n + 1$ is a submanifold of dimension $n$ that is everywhere tangent to the contact hyperplanes. Legendrian submanifolds play in contact geometry the role that Lagrangian submanifolds play in symplectic geometry. In a contact $3$-manifold they are Legendrian knots, whose classification up to Legendrian isotopy is a rich subject with invariants such as the Thurston–Bennequin number.