on a symplectic manifold take on the canonical form, this example suggests that Lagrangian submanifolds are relatively unconstrained. The classification of symplectic manifolds is done via Floer homology—this is an application of Morse theory to the action functional for maps between Lagrangian submanifolds. In physics, the action describes the time evolution of a physical system; here, it can be taken as the description of the dynamics of branes. Example: Morse theory Another useful class of Lagrangian submanifolds occur in Morse theory. Given a Morse function and for a small enough one can construct a Lagrangian submanifold given by the vanishing locus . For a generic Morse function we have a Lagrangian intersection given by . Special Lagrangian submanifolds In the case of Kahler manifolds (or Calabi–Yau manifolds) we can make a choice on as a holomorphic n-form, where is the real part and imaginary. A Lagrangian submanifold is called special if in addition to the above Lagrangian condition the restriction to is vanishing. In other words, the real part restricted on leads the volume form on . The following examples are known as special Lagrangian submanifolds, complex Lagrangian submanifolds of hyperKahler manifolds, fixed points of a real structure of Calabi–Yau manifolds. The SYZ conjecture deals with the study of special Lagrangian submanifolds in mirror symmetry; see . The Thomas–Yau conjecture predicts that the existence of a special Lagrangian submanifolds on Calabi–Yau manifolds in Hamiltonian isotopy classes of Lagrangians is equivalent to stability with respect to a stability condition on the Fukaya category of the manifold. Lagrangian fibration A Lagrangian fibration of a symplectic manifold M is a fibration where all of the fibres are Lagrangian submanifolds. Since M is even-dimensional we can take local coordinates and by Darboux's theorem the symplectic form ω can be, at least locally, written as , where d denotes the exterior derivative and ∧ denotes the exterior product. This form is called the Poincaré two-form or the canonical two-form. Using this set-up we can locally think of M as being the cotangent bundle and the Lagrangian fibration as the trivial fibration This is the canonical picture. Lagrangian mapping Let L be a Lagrangian submanifold of a symplectic manifold (K,ω) given by an immersion (i is called a Lagrangian immersion). Let give a Lagrangian fibration of K. The composite is a Lagrangian mapping. The critical value set of π ∘ i is called a caustic. Two