## 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)

### MSC:

 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
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### References:

 [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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