\(R^{2n}\) is a universal symplectic manifold for reduction
The authors show that under a mild topological restriction, all symplectic manifolds are obtained by reduction from \({\mathbb{R}}^{2n}\) with its standard symplectic structure. Reduction is a well-known construction in symplectic geometry to get a symplectic manifold from some submanifold of a given symplectic manifold. See for example \textit{R. Abraham} and \textit{J. E. Marsden} [Foundations of mechanics, 2nd ed. (New York 1978; Zbl 0393.70001)]. More precisely, if (M,\(\omega)\) is a symplectic manifold we consider a submanifold N such that the pullback \(\omega_ N\) of \(\omega\) to N has constant rank. Let ker \(\omega_ N\) be the foliation defined by the characteristic distribution of \(\omega_ N\). If we assume further that \(M_ N=N/\ker \omega_ N\) is a manifold and \(N\to M_ N\) is a submersion, it inherits a symplectic structure \((\omega_ N)\). We say that \(M_ N\) is the reduction of M by N. An important particular case is the so-called Marsden-Weinstein reduction, when there is a symplectic action by a Lie group G on M, and N is taken as a level set for the corresponding moment of the action. The authors prove in a slightly more general context that every symplectic manifold \((Q,\Omega)\) with Q of finite type can be realized as a reduction of some \({\mathbb{R}}^{2n}\). The finite type condition is satisfied by all well-behaved manifolds. Their other main results answer partially the question of when a symplectic manifold arises from a Marsden-Weinstein reduction of \({\mathbb{R}}^{2n}\) with standard symplectic structure, relative to a Hamiltonian action of a connected abelian group. Non canonical bundles (comprising the canonical cotangent bundles) are always realized in this way. However, determining which symplectic manifolds arise as a Marsden-Weinstein reduction of some \({\mathbb{R}}^{2n}\) is an apparently difficult open problem.
scientific article; zbMATH DE number 4131189