A universal model for cosymplectic manifolds. (English) Zbl 0861.53026

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 \(K=\text{rank}(\eta)\).
Reviewer: R.Roşca (Paris)


53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
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[1] Albert, C., Le thèoreme de réduction de Marsden-Weinstein en géométrie cosymplectique et de contact, J. Geom. Phys., 6, 627-649 (1989) · Zbl 0712.53017
[2] Cantrijn, F.; de León, M.; Lacomba, E. A., Gradient vector fields on cosymplectic manifolds, J. Phys. A: Math. Gen., 25, 175-188 (1992) · Zbl 0754.53024
[3] Gotay, M. J.; Tuynman, G. M., \(R^{2n}\) is a universal symplectic manifold for reduction, Lett. Math. Phys., 18, 55-59 (1989) · Zbl 0691.53019
[4] Gotay, M. J.; Tuynman, G. M., A symplectic analogue of the Mostow-Palais theorem symplectic geometry, groupoids and integrable systems, (Dazord, P.; Weinstein, A., MSRI Publications, Vol. 20 (1991), Springer: Springer Berlin), 173-182 · Zbl 0737.58022
[5] de León, M.; Rodrigues, P. R., Methods of differential geometry in analytical mechanics, (Math. Ser., Vol. 152 (1989), North-Holand: North-Holand Amsterdam) · Zbl 0757.58014
[6] Libermann, P.; Marle, Ch.-M., Symplectic Geometry and Analytical Mechanics (1987), Reidel: Reidel Dordrecht
[7] de Léon, M.; Saralegui, M., Cosymplectic reduction for singular momentum maps, J. Phys. A: Math. Gen., 26, 5033-5043 (1993) · Zbl 0806.53033
[8] Whitehead, G. W., Elements of Homotopy Theory (1978), Springer: Springer Berlin · Zbl 0151.31101
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