A singular Poincaré lemma. (English) Zbl 1078.58007

Miranda, Eva; Vũ Ngọc, San

Int. Math. Res. Not. 2005, No. 1, 27-46 (2005).

The authors prove the following result: let \(g_{1},\dots , g_{r}\) be a set of germs of smooth functions on \((\mathbb{R}^{2n}, 0), r\leq n\) fulfilling the commutation relations: \(X_{i}(g_{j})=X_{j}(g_{i})\) for any \(i,j \in \{ 1,2,\dots ,r \}\), where \(\{ X_{1},\dots , X_{r} \}\) is a basis of a Cartan subalgebra of \(\text{sp}(2n, \mathbb{R})\). Then there exists a germ of a smooth function \(G\) and \(r\) germs of smooth functions \(f_{i}\) such that: (i) \(X_{j}(f_{i})=0\) for any \(i,j \in \{ 1,2,\dots ,r \}\); (ii) \(g_{i}=f_{i} + X_{i}(G)\) for any \(i \in \{ 1,2,\dots ,r \}\).
Some applications are also nicely pointed out.

Reviewer: Mircea Puta (Timişoara)

MSC:

58C25	Differentiable maps on manifolds
37J35	Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
53D05	Symplectic manifolds, general
58A10	Differential forms in global analysis

Keywords:

singular Poincaré lemma; Cartan subalgebra