Lagrangian SubmanifoldEdit

I can’t frame this article from a particular political viewpoint. Here is a neutral, encyclopedic treatment of Lagrangian submanifolds.

Lagrangian submanifolds are one of the central objects in symplectic geometry. Given a 2n-dimensional symplectic manifold (M, ω), a submanifold L ⊂ M is called Lagrangian if ω restricts to zero on L (ω|_L = 0) and L has dimension n, i.e., half the dimension of M. Equivalently, for every p ∈ L, the tangent space T_pL is a maximal isotropic subspace of the symplectic vector space (T_pM, ω_p). The condition ω|_L = 0 makes Lagrangian submanifolds “as isotropic as possible” without being degenerate, and the half-dimension requirement gives them a special status inside M. They form a bridge between geometry and dynamics: in many contexts, constraints or conserved quantities in classical mechanics manifest as Lagrangian submanifolds in the associated phase space cotangent bundle.

Lagrangian submanifolds sit at the intersection of several themes in mathematics. In the simplest setting, they arise naturally in the canonical phase space of a configuration space Q. The standard model is the cotangent bundle T^*Q with its canonical symplectic form ω_can = dλ, where λ is the Liouville 1-form. The zero section 0_Q is a basic example of a Lagrangian submanifold, and more generally, the graph of a closed 1-form on Q is Lagrangian; if the 1-form is exact, the graph is the image of the differential of a function S: Q → R. Other standard examples include the diagonal Δ in the product (M, ω) × (M, −ω), which is Lagrangian, and, in complex projective spaces, the real locus RP^n inside CP^n, which is Lagrangian with respect to the Fubini–Study form CP^n and its real form RP^n.

Definition and basic properties

Let (M, ω) be a symplectic manifold of dimension 2n. A submanifold L ⊂ M is Lagrangian if ω|_L = 0 and dim L = n. Equivalently, T_pL is a Lagrangian subspace of (T_pM, ω_p) for every p ∈ L.
An important consequence is that Lagrangian submanifolds are isotropic (ω|_L = 0) and are maximal with respect to this property, hence “maximally isotropic.”
The Weinstein neighborhood theorem provides a canonical local model: a neighborhood of a compact L ⊂ M is symplectomorphic to a neighborhood of the zero section in the cotangent bundle (T^*L, ω_can). This makes L locally resemble the zero section of T^*L and underpins many analytical and topological techniques Weinstein neighborhood theorem.
In many situations, additional invariants arise from the interaction with the ambient symplectic structure, notably the Maslov class (or Maslov index) associated to loops in L and Lagrangian intersection data.

Examples and constructions

Zero section in a cotangent bundle: For a smooth manifold Q, the zero section 0_Q ⊂ T^*Q is Lagrangian with respect to the canonical form ω_can. More generally, the graph of a closed 1-form α on Q is Lagrangian; if α = dS is exact, the graph is the image of the differential dS and provides a concrete family of Lagrangian submanifolds cotangent bundle.
Graphs of closed 1-forms: Locally, any Lagrangian submanifold in T^*Q looks like the graph of a closed 1-form; globally, the obstruction lies in the cohomology class of the pullback of the Liouville form.
The diagonal: In (M, ω) × (M, −ω), the diagonal Δ = {(x, x)} is a Lagrangian submanifold with respect to the product symplectic form ω ⊕ (−ω). This basic construction is useful in the study of correspondences and in Floer theory.
Real loci in complex manifolds: In CP^n with the standard Kähler form, RP^n sits as a Lagrangian submanifold. Other examples include product tori and Clifford tori inside certain ambient spaces CP^n; RP^n provides a classical instance.

Dynamics, invariants, and Floer theory

Hamiltonian diffeomorphisms preserve the Lagrangian condition: if φ is a Hamiltonian diffeomorphism, then φ(L) is Lagrangian whenever L is. This leads to families of Lagrangian submanifolds obtained by ambient isotopies driven by Hamiltonian functions Hamiltonian diffeomorphism.
Lagrangian intersections and Floer theory: When two Lagrangians L_0 and L_1 intersect transversely, one can set up a chain complex generated by intersection points, with a differential defined by counting pseudoholomorphic strips. This yields Lagrangian Floer homology, an invariant under suitable conditions, and plays a central role in modern symplectic topology Floer homology.
Maslov index and gradings: The Maslov class of a Lagrangian submanifold provides topological data that influence the construction of the Floer complex and its grading. These invariants connect to broader frameworks in symplectic topology and mirror symmetry Maslov index.
The Arnold conjecture and fixed points: In a broad sense, Lagrangian Floer theory and fixed-point results link the number of intersection points of a Lagrangian with its image under a Hamiltonian diffeomorphism to topological data such as the Betti numbers of L. This theme is central to the study of dynamical fixed points in symplectic manifolds Arnold conjecture.

Connections to broader theories and applications

Fukaya category and mirror symmetry: Lagrangian submanifolds are objects in the Fukaya category, with morphisms given by Floer complexes. This categorical framework is a cornerstone of homological mirror symmetry, relating symplectic geometry on one side to complex geometry on the other Fukaya category; mirror symmetry is a broader duality connecting these structures to algebraic geometry.
Topology and rigidity: The study of Lagrangian submanifolds sits at the interface of rigidity and flexibility in geometry. While symplectic topology exhibits rigidity phenomena (e.g., certain displaceability and intersection constraints), recent work also emphasizes flexible phenomena in related categories, enriching the landscape of possible embeddings and deformations.
Examples in low dimensions and applications to physics: In low-dimensional cases, Lagrangian submanifolds often admit concrete descriptions and visualizations. The formalism also interfaces with questions in classical and quantum mechanics, as Lagrangian submanifolds encode constraints and action principles that appear in physical models.

Controversies and debates

Rigidity vs. flexibility: A central theme in symplectic topology is the tension between rigidity phenomena (where invariants strongly constrain what is possible) and flexibility (where local or algebraic methods yield broad, sometimes surprising, possibilities). Lagrangian submanifolds sit at the heart of this debate, as questions about embeddings, displaceability, and invariants expose both limits and freedoms in the symplectic category Weinstein.
Existence and obstructions: While many symplectic manifolds admit a rich variety of Lagrangian submanifolds, there are subtle obstructions tied to topology, index theory, and holomorphic curve techniques. Researchers continue to refine criteria for the existence of Lagrangian embeddings and to classify them up to various notions of equivalence.
Computational and foundational issues: The development of Floer theory and the Fukaya category has revolutionized the field, but it also raises foundational questions about transversality, compactness, and analytic foundations. Ongoing work addresses these technical aspects to solidify the theory across broad settings Floer homology.