Weinstein's neighborhood theorem
/ WINE-stine /
Darboux's theorem says that near a point, every symplectic manifold looks standard. Weinstein's theorem upgrades this from a point to a whole Lagrangian submanifold: it says that a neighborhood of any Lagrangian L looks exactly like a neighborhood of the zero section inside the cotangent bundle T*L. In other words, the only data that determine how a Lagrangian sits in its symplectic surroundings — up to symplectomorphism — is the intrinsic manifold L itself. The embedding has no local secrets beyond L's own diffeomorphism type.
Precisely, Weinstein's Lagrangian neighborhood theorem states: if L is a Lagrangian submanifold of (M, omega), then there is a symplectomorphism from a neighborhood of L in M onto a neighborhood of the zero section in (T*L, omega_can), carrying L to the zero section. The proof is again the Moser trick: build a diffeomorphism matching L to the zero section, show the two symplectic forms agree along L, interpolate, and flow the difference away using closedness. There is a companion isotropic version (Weinstein's isotropic neighborhood theorem) where the model is a bundle built from the symplectic normal data, and an analogous statement that two symplectomorphic-on-the-nose Lagrangian embeddings are equivalent near L.
Why it matters: this is the foundational rigidity-and-flexibility result for Lagrangians. It means deformations of a Lagrangian L are governed by closed 1-forms on L (graphs of which are the nearby Lagrangians), so the local theory of Lagrangians reduces to the de Rham cohomology H^1(L; R) — exact deformations are Hamiltonian, the rest are 'flux'. It is the starting point for Floer theory, Lagrangian surgery, and the whole 'symplectic manifolds are built from cotangent pieces' philosophy. A caveat worth stating: the theorem is purely about a neighborhood (a germ); it says nothing about whether a given abstract L can be embedded as a Lagrangian in a given M at all — global Lagrangian embedding problems (e.g. which manifolds embed Lagrangianly in C^n) are deep and largely open.
Take L = S^1, a circle, sitting as a Lagrangian inside some symplectic surface. Weinstein says a neighborhood of that circle is symplectomorphic to a neighborhood of the zero section of T*S^1 = S^1 x R, the cylinder with omega = dp ^ dtheta. Nearby Lagrangian circles correspond to graphs of closed 1-forms on S^1; since H^1(S^1; R) = R, there is a one-parameter family of non-isotopic-through-exact deformations, parametrized by the flux integral of the 1-form around the circle.
A Lagrangian circle's neighborhood is modeled on T*S^1; deformations are governed by H^1(S^1) = R.
The theorem is local (a germ around L). It does NOT solve the global existence question of whether L embeds as a Lagrangian in a given M — that is a separate, often hard, problem (Gromov's results forbid, e.g., an exact Lagrangian sphere in C^n).