Coisotropic submanifold - Imported Knowlpedia Content

Definition

Let $(M,\omega)$ be a symplectic manifold and $C\subseteq M$ an embedded submanifold. For $p\in C$ , let

(T_pC)^\omega=\{v\in T_pM:\omega_p(v,w)=0\text{ for all }w\in T_pC\}.

The submanifold $C$ is coisotropic if

(T_pC)^\omega\subseteq T_pC

at every $p\in C$ ; equivalently, every $T_pC$ is a coisotropic subspace of $T_pM$ . The kernel of the restricted form $\omega|_{T_pC}$ is exactly $(T_pC)^\omega$ . Thus coisotropy says that all null directions of the restricted form are tangent to $C$ . It depends on the ambient form, not only on the smooth embedding.

Dimension and local structure

If $\dim M=2n$ and $C$ has codimension $k$ , then the characteristic space $(T_pC)^\omega$ has dimension $k$ . Coisotropy forces $k\leq n$ , or equivalently $\dim C\geq n$ . A hypersurface in a symplectic manifold is automatically coisotropic because its one-dimensional symplectic orthogonal lies in the hyperplane.

The spaces $(T_pC)^\omega$ assemble into the characteristic distribution on $C$ . Since the restricted two-form is closed, this distribution is involutive and hence determines the characteristic foliation.

Examples and contrasts

The whole manifold $M$ is coisotropic. Every Lagrangian submanifold is coisotropic because its tangent spaces equal their symplectic orthogonals. If a Hamiltonian Lie-group action has moment map $\mu:M\to\mathfrak g^*$ , then a regular level set $\mu^{-1}(\xi)$ is coisotropic under the usual hypotheses when $\xi$ is fixed by the coadjoint action relevant to the reduction.

A proper symplectic submanifold is not coisotropic: the restricted form has zero kernel, whereas a positive-codimension coisotropic submanifold has a positive-dimensional characteristic space.

Role in reduction

The leaf space of the characteristic foliation is the candidate reduced phase space. When that leaf space is a smooth manifold and the quotient map is a submersion, there is a unique symplectic form downstairs whose pullback is $\omega|_C$ . Without these regularity hypotheses, the quotient may be singular or non-Hausdorff, even though $C$ is a perfectly smooth coisotropic submanifold.

References

Ana Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics 1764, Springer, 2008. Springer DOI record. Relevant: symplectic reduction and the characteristic directions of constraint sets.
Victor Guillemin and Shlomo Sternberg, Symplectic Techniques in Physics, Cambridge University Press, 1984. Cambridge DOI record. Relevant: Chapter 5, constraints and symplectic reduction.