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Coisotropic submanifolds and dual pairs. (English) Zbl 1326.53119
Consider a finite-dimensional manifold \(M\). Denote by \(I\) the unit interval \([0, 1]\). The space of differentiable paths on \(M\) is denoted by \(P M := C^1 (I, M)\), and the space of bundle maps \( T I \to T^*M\) with continuously differentiable base map and continuous fiber map is denoted by \(T^*P M\). The space \(P M\) is considered as a Banach manifold and \(T^*P M\) as a Banach, weak symplectic manifold (a weak symplectic form is a closed 2-form that induces an injective map from the tangent to the cotangent bundle).
Fix a section \(\pi\) in \(TM \otimes TM\), and consider its induced bundle map \(\pi^{\sharp}: T^*M \to TM\). For two given submanifolds \(C_{0}\) and \(C_{1}\) of \(M\), denoted by \(\mathcal{C}_{\pi}(M; C_0,C_1)\), the subpace of \(T^*PM\) of “\(\pi\)-compatible path” from \(C_0\) to \(C_1\), and by \(p_i: \mathcal{C}_{\pi}(M; C_0,C_1) \to C_i\), \(i=0,1\), the canonical maps. Contrary to the case when \(C_0=C_1=M\), \(\mathcal{C}_{\pi}(M; C_0,C_1)\) is not in general a Banach submanifold. Although, \(\mathcal{C}_{\pi}(M; C_0,C_1)\) is not a submanifold, one still can define the coisotropy criteria, using the notion of tangent bundle with fibers being the Zarisky tangent space at each point. Specifically, in the symplectic case, a subvariety (i.e., a common zero set of a family of smooth functions) is said to be coisotropic when its symplectic orthogonal bundle is contained in its tangent bundle.
The first main result of the paper says that \(\mathcal{C}_{\pi}(M; M,M)\) is coisotropic in \(T^*PM\) if and only if, \(\pi\) is a Poisson bivector field. The later means that \(\pi\) is a skew-symmetric 2-tensor field satisfying \([\pi,\pi]=0\), where \([\,,\,]\) is the Schouten-Nijenhuis bracket. The second main result asserts that, if \(\pi\) is a Poisson bivector field, then \(\mathcal{C}_{\pi}(M; C_0,C_1)\) is coisotropic in \(T^*PM\) if and only if the subsets \(\mathrm{Im}(p_0)\) and \(\mathrm{Im}(p_1)\) are coisotropic in \(M\) relative to \(C_0\) and \(C_1\) respectively, i.e., \(\pi^{\sharp}(N^*_xC_i) \subseteq T_xC_i\), for all \(x \in \mathrm{Im}(p_i)\), \(i=0,1\).

MSC:
53D17 Poisson manifolds; Poisson groupoids and algebroids
81T45 Topological field theories in quantum mechanics
53D20 Momentum maps; symplectic reduction
58H05 Pseudogroups and differentiable groupoids
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