A universal model for cosymplectic manifolds
A cosymplectic manifold is a triple \((M,\Omega,\eta)\) consisting of a \(C^\infty(2n+1)\)-dimensional manifold \(M\), endowed with a closed 2-form \(\Omega\) and a closed 1-form \(\eta\) such that \(\Omega^n \wedge \eta \neq 0\) everywhere. Let \(b:TM\to T^*M\), \(X \in TM\mapsto i_X\Omega+\eta(X)\eta\) be the bundle morphism and \(R=b^{-1}(\eta)\) the Reeb vector field, i.e. \(i_R\Omega =0\), \(\eta(R)=1\). Two cosymplectic manifolds \((M_1,\Omega_1,\eta_1)\) and \((M_2,\Omega_2,\eta_2)\) are said to be isomorphic if there exists a diffeomorphism \(\Phi:M_1\to M_2\) such that \(\Phi^* \Omega_2=\Omega_1\), \(\Phi^* \eta_2=\eta_1\). If \(C\) is a submanifold of \(M\), the authors assume that the following conditions are satisfied: (i) \(R\) is tangent to \(C\); (ii) the characteristic distribution \(F=\text{ker }\Omega|_C \cap \text{ker }\eta|_C\) is a foliation on \(C\); (iii) the space of leaves \(M=C/F\) has the structure of a manifold and the canonical projection \(\pi:C\to M_p\) is a fibration. With these hypotheses it is shown that there exist unique closed forms \(\Omega_p\) and \(\eta_p\) on \(M_p\), such that: (a) \(\pi^* \Omega_p=\Omega|_C\) and \(\pi^*\eta_p=\eta|_C\); (b) \((M_p,\Omega_p,\eta_p)\) is a cosymplectic manifold; (c) \(\pi_*\) \((R|_C)=R_p\), where \(R_p\) is the Reeb vector field of \(M_p\). Under these circumstances \(M_p\) is said to be the reduction of \(M\) by \(C\). Since the local model for a cosymplectic manifold is \(\mathbb{R}^{2n+1}\) with the 2-form \(d\theta_{\mathbb{R}^n}\) and the 1-form \(ds\), the following main theorem is formulated: Let \((M,\Omega,\eta)\) be a cosymplectic manifold of finite type. Then there exist integers \(N\) and \(K\) and real numbers \(\mu_1,\dots,\mu_K\) that are independent over \(\mathbb{Q}\), such that \(M\) is the reduction of the cosymplectic manifold \((M_u,\Omega_u,\eta_u)\) by some \(C \subset M_u\), where \[ M_u=\mathbb{R}\times T^k(\mathbb{T}^*\times \mathbb{R}^N),\quad \Omega_u=d\theta_{\mathbb{T}^K\times \mathbb{R}^N},\quad \eta_u=ds+\sum^K_{i=1} \mu_i d\varphi_i \] with \(\varphi_i\) the angle coordinates on the torus \(\mathbb{T}^K\), \(s\) the canonical coordinate in \(\mathbb{R}\), and \(K=\text{rank}(\eta)\).
scientific article; zbMATH DE number 936183