Knowlpedia

Theorem

Coisotropic reduction

A smooth quotient of a coisotropic submanifold by its characteristic leaves inherits a unique symplectic form.

Statement

Let CC be a of (M,ω)(M,\omega), with inclusion i:C↪Mi:C\hookrightarrow M. Suppose the leaf space C/KC/\mathcal K of its is a BB, and the quotient map π:C→B\pi:C\to B is a surjective submersion whose fibers are the characteristic leaves. Then there is a unique two-form ωB\omega_B satisfying

π∗ωB=i∗ω.\pi^*\omega_B=i^*\omega.

This form is closed and nondegenerate, so (B,ωB)(B,\omega_B) is a . It is the coisotropic reduction of CC. Smoothness and the submersion condition are hypotheses, not consequences of coisotropy.

Descent and nondegeneracy

The kernel of i∗ωi^*\omega is exactly K\mathcal K. The form is horizontal because it annihilates every vector tangent to a leaf. It is also invariant along leafwise : if XX is tangent to K\mathcal K, then Cartan's formula gives

LX(i∗ω)=dιX(i∗ω)+ιXd(i∗ω)=0.\mathcal L_X(i^*\omega)=d\iota_X(i^*\omega)+\iota_Xd(i^*\omega)=0.

Thus i∗ωi^*\omega is basic and descends uniquely through the surjective submersion π\pi. If a tangent vector downstairs pairs to zero with every other tangent vector, any lift lies in ker⁡i∗ω=ker⁡dπ\ker i^*\omega=\ker d\pi, so the original vector is zero. This proves nondegeneracy Cannas da Silva, §1.4.

Linear model and Hamiltonian reduction

For a WW of a , the theorem becomes :

Wred=W/Wω.W_{\mathrm{red}}=W/W^\omega.

There are no global leaf-space pathologies in this linear case.

Regular Hamiltonian reduction is the principal nonlinear example. Under the hypotheses of the , the moment-map level set is coisotropic, its characteristic leaves are group orbits, and its coisotropic reduction is the usual .

Failure of the hypotheses

The characteristic foliation may have dense leaves, nonclosed leaves, or a non-Hausdorff leaf space. Even when the underlying quotient is Hausdorff, it may not admit a smooth structure for which π\pi is a submersion. In these cases the pullback equation still describes the desired geometry formally, but it does not produce an ordinary symplectic manifold.

References
  1. Ana Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics 1764, Springer, 2001. DOI record. Relevant: §1.4, characteristic foliations and symplectic reduction.
  2. Paulette Libermann and Charles-Michel Marle, Symplectic Geometry and Analytical Mechanics, Mathematics and Its Applications 35, D. Reidel, 1987. DOI record. Relevant: Chapter III, reduction of presymplectic and coisotropic manifolds.