Knowlpedia

Theorem

Lagrangian neighborhood theorem

A neighborhood of a Lagrangian submanifold has the cotangent bundle as its canonical local symplectic model.

Statement

Let LL be an embedded of a (M,ω)(M,\omega). There are neighborhoods UU of the in the T∗LT^*L and VV of LL in MM, together with a

Φ:(U,ωcan)⟶(V,ω)\Phi:(U,\omega_{\mathrm{can}})\longrightarrow(V,\omega)

whose restriction to the zero section is the given inclusion L↪ML\hookrightarrow M. Thus the germ of the ambient symplectic structure along LL is modeled by the canonical . The neighborhoods and the symplectomorphism are generally not unique, and the theorem makes no global claim about all of T∗LT^*L or MM.

Relative formulation

A useful equivalent form compares two symplectic manifolds containing copies of the same Lagrangian LL. If a diffeomorphism between those copies is fixed, then it extends, after shrinking neighborhoods, to a symplectomorphism near LL. Taking one ambient manifold to be T∗LT^*L and one copy to be its zero section gives the stated form. This is the Lagrangian case of Weinstein's neighborhood results Weinstein, Theorem 6.1.

Proof idea

The symplectic form identifies the TM∣L/TLTM|_L/TL with T∗LT^*L: the class of vv is sent to the covector w↦ω(v,w)w\mapsto\omega(v,w). A tubular-neighborhood choice realizes this bundle identification by a diffeomorphism near the zero section. The pulled-back ambient form and the canonical cotangent form agree along LL. A relative Moser argument then corrects the diffeomorphism to a symplectomorphism while fixing LL.

The correction step explains both the local nature of the theorem and its nonuniqueness: auxiliary tubular data and the Moser primitive can vary.

Consequences and limitations

Nearby sections of T∗LT^*L are Lagrangian exactly when their defining one-forms are closed. Consequently, sufficiently small Lagrangian deformations of LL, once placed in a Weinstein neighborhood and transverse to the cotangent fibers, are represented by closed one-forms. Exact one-forms describe the corresponding local Hamiltonian deformations.

References
  1. Alan Weinstein, “Symplectic Manifolds and Their Lagrangian Submanifolds,” Advances in Mathematics 6 (1971), 329–346. DOI record. Relevant: Theorem 6.1 and the neighborhood-equivalence argument.
  2. Dusa McDuff and Dietmar Salamon, Introduction to Symplectic Topology, 3rd ed., Oxford University Press, 2017. DOI record. Relevant: §3.4, the Weinstein neighborhood theorem and local Lagrangian graphs.